DOCSLIB.ORG

Amalie (Emmy) Noether (1882 - 1935)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Amalie (Emmy) Noether (1882 - 1935) Mairi Sakellariadou King’s College London Emmy Noether was born in Erlangen, Germany on March 23, 1882 She was named Amalie, but always called "Emmy" 2 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The family The father: Max Noether (1844 Mannheim– 1921 Erlangen) From a Jewish family of wealthy wholesale hardware dealers. At 14, Max contracted polio and was afflicted by its effects for the rest of his life. Through self-study, he learned advanced mathematics and entered the University of Heidelberg in 1865. He moved to the University of Erlagen in 1888. While there, he helped to found the field of algebraic geometry. In 1880 he married Ida Amalia Kaufmann, the daughter of another wealthy Jewish merchant family. Two years later they had their first child, named Amalia (“ Emmy “) after her mother. 3 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The family The mother: Ida Amalia Kaufmann (1852 Koln – 1951 Erlagen) From a wealthy Jewish merchant family. Ida had a brother who was a professor at the University of Berlin. In 1880 she married Max Noether ; they had four children. Ida was a skilled pianist. 4 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The family . Emmy 1882 – 1935 Professor of Mathematics in Erlagen, Gottingen, and Bryn Mawr (USA) . Alfred 1883 – 1918 Chemist . Fritz 1884 – 1937 Professor of Mathematics in Breslavia (Germany) and in Tomsk (Russia) . Gustav Robert 1889 -- 1928 5 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The family a bit before the first world war 6 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Erlagen period (1882 – 1915) Emmy's childhood was unexceptional, going to school, learning domestic skills, and taking piano lessons. Her passion was dancing. Since girls were not eligible to enroll in the gymnasium, she attended the Municipal School for Higher Education of Daughters in Erlangen, where she studied arithmetic and languages. Emmy also loved mathematics, but she knew that the rules of the time meant she would not be allowed to follow in her father’s footsteps to become a University academic. At age18, she was qualified to teach English and French in girls’ schools. 7 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Erlagen period (1882 – 1915) Although a career in teaching offered her financial security, her love of mathematics proved to be too strong. Emmy decided to abandon teaching and apply to the University of Erlangen to observe mathematics lectures. She could only observe lectures, because women were not permitted to enroll officially at the University. Emmy was one of the two female students sitting in on courses at Erlangen. Between 1900 and 1902 Emmy studied mathematics at Erlangen. In July 1903 she went to Nürnberg and passed the matriculation examination allowing her to study mathematics (but not officially enroll) at any German University. 8 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Erlagen period (1882 – 1915) Emmy chose to go for a semester to the University of Göttingen. She attended lectures given by: Schwarzschild Minkowski Blumenthal Again she was not allowed to be a properly matriculated student but was only allowed to sit in on lectures. After one semester at Göttingen, Emmy returned to Erlangen. Klein Hilbert 9 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Erlagen period (1882 – 1915) At this point the rules were changed and women students were allowed to matriculate on an equal basis to the men. On 24 October 1904 Emmy matriculated at Erlangen and in 1907, at the age of 25, she was granted a doctorate after working under Paul Gordan,. Her thesis was entitled “On the construction of the system of forms of a ternary biquadratic form “ (the search for the invariants of a homogeneous polynomial of degree 4 in 3 variables). Emmy was the only student Gordan ever accepted as a Ph.D. candidate. “.. her dissertation with Gordan pursued a huge calculation that had stumped Gordan forty years before and which Noether could not complete either. So far as I know no one has ever completed it or even checked it as far as she went. “ Colin McLarty (2011) 10 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Erlagen period (1882 – 1915) Research and teaching at Erlagen University (1908-1915) Having completed her doctorate the normal progression to an academic post would have been the habilitation. However this route was not open to women so Emmy remained at Erlangen, helping her father who, particularly because of his own disabilities, was grateful for his daughter's help. Emmy also worked on her own research; she was influenced by Ernst Fischer who had succeeded Gordan to the chair of mathematics when he retired in 1911. Emmy wrote about Fischer's influence: “Above all I am indebted to Mr E Fischer from whom I received the decisive impulse to study abstract algebra from an arithmetical viewpoint, and this remained the governing idea for all my later work. “ 11 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou Research and teaching at Erlagen University (1908-1915) Dr. Noether, Mathematics Lecturer In 1908 Emmy was appointed to the position of mathematics lecturer at Erlangen. Unfortunately, it was an unpaid position. Emmy’s parents supported her as much as they could through this time. Nevertheless, her life was a struggle financially. While working as a lecturer, Emmy became fascinated by work Hilbert had done in Göttingen. 1908: member of the Mathematical Circle of Palermo . 1909: member of the Mathematical German Society 12 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Göttingen period (1915 –1933) Hilbert was working on physics, in particular on ideas on the theory of relativity close to those of Albert Einstein. He decided that he needed the help of an expert on invariant theory and, after discussions with Klein, they issued the invitation. Felix Klein 1849 – 1925 David Hilbert: 1862 – 1943 13 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Göttingen period (1915 –1933) In 1915 Hilbert invited her to become a lecturer in Göttingen. This provoked a storm of protest from philologists and historians among the faculty. One faculty member protested: ‘’What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman? ‘’ Hilbert responded with indignation, stating, “ I do not see that the sex of the candidate is an argument against her admission ... After all, we are a university, not a bath house. ‘’ Emmy was so eager to join Hilbert’s department in Göttingen that, to overcome Hilbert’s opponents, she agreed not to be formally appointed as a lecturer and to receive no pay. Her father continued supporting her financially (her mother died in 1915) and the lectures she gave were advertised as lectures by Professor Hilbert, with assistance from Dr. E. Noether. 14 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Göttingen period (1915 –1933) Soon after arriving at Göttingen, Noether proved her two theorems in 1915, published in 1918, under the title Invariante Variationsprobleme 15 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Göttingen period (1915 –1933) Soon after arriving at Göttingen, Noether proved her two theorems in 1915, published in 1918, under the title Invariante Variationsprobleme in Nachrichten von der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-physikalische Klasse, 1918, pp. 235-257. 16 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Göttingen period (1915 –1933) Soon after arriving at Göttingen, Noether proved her two theorems in 1915, published in 1918, under the title Invariante Variationsprobleme in Nachrichten von der Koniglichen Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-physikalische Klasse, 1918, pp. 235-257. Emmy’s theorems relate symmetry groups of a variational integral to properties of its associated Euler-Lagrange equations. Every differentiable symmetry of the action of a physical system has a corresponding conservation law. Among the most important mathematical theorems ever proved in guiding the development of modern physics. 17 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou Emmy submitted the “invariant Variationsprobleme” for her Habilitation, finally obtained in 1919. She never referred to her article in her subsequent publications. In Göttingen, Emmy had only one immediate follower, Erich Bessel-Hagen (1898-1946), who was Klein’s student. He formulated the two Noether theorems slightly more general than they had been formulated in her article, and added “I owe these to an oral communication by Miss Emmy Noether herself “. 18 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Göttingen period (1915 –1933) November 1915 Albert Einstein publishes his theory of General Relativity. David Hilbert states the Variational Principle. The contribution of Emmy’s work was fundamental. Albert Einstein: 1879-1955 David Hilbert: 1862 – 1943 19 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou Although Noether's theorem had a profound effect upon physics, among mathematicians she is best remembered for her seminal contributions to Abstract Algebra. In 1924 B. L. van der Waerden, arrived at the University of Göttingen. van der Waerden later said that her originality was “absolute beyond comparison “. In 1931 van der Waerden published Modern Algebra, a central text in the field; its second volume borrowed heavily from Emmy's work. “ ... The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries. .." Nathan Jacobson in his introduction to Nother’s collected papers 20 Emmy Noether 10th February 2016 Zakopane Mairi Sakellariadou The Göttingen period (1922 –1933) Assistant professor in 1922 During her time at the University of Gottingen, she accumulated a small following of students known as Noether's boys.
Recommended publications
  • A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol

    A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol

    Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge­ bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe­ maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe­ matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members.
  • Mathematicians Fleeing from Nazi Germany

    Mathematicians Fleeing from Nazi Germany

    Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D.
  • Algebraic Topology - Wikipedia, the Free Encyclopedia Page 1 of 5

    Algebraic Topology - Wikipedia, the Free Encyclopedia Page 1 of 5

    Algebraic topology - Wikipedia, the free encyclopedia Page 1 of 5 Algebraic topology From Wikipedia, the free encyclopedia Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Contents 1 The method of algebraic invariants 2 Setting in category theory 3 Results on homology 4 Applications of algebraic topology 5 Notable algebraic topologists 6 Important theorems in algebraic topology 7 See also 8 Notes 9 References 10 Further reading The method of algebraic invariants An older name for the subject was combinatorial topology , implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW-complex ). The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups.
  • Emmy Noether, Greatest Woman Mathematician Clark Kimberling

    Emmy Noether, Greatest Woman Mathematician Clark Kimberling

    Emmy Noether, Greatest Woman Mathematician Clark Kimberling Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). With more than 100,000 members, NCTM is the largest organization dedicated to the improvement of mathematics education and to the needs of teachers of mathematics. Founded in 1920 as a not-for-profit professional and educational association, NCTM has opened doors to vast sources of publications, products, and services to help teachers do a better job in the classroom. For more information on membership in the NCTM, call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright March 1982 by the National Council of Teachers of Mathematics. All rights reserved. mmy Noether was born over one hundred years ago in the German university town of Erlangen, where her father, Max Noether, was a professor of Emathematics. At that time it was very unusual for a woman to seek a university education. In fact, a leading historian of the day wrote that talk of “surrendering our universities to the invasion of women . is a shameful display of moral weakness.”1 At the University of Erlangen, the Academic Senate in 1898 declared that the admission of women students would “overthrow all academic order.”2 In spite of all this, Emmy Noether was able to attend lectures at Erlangen in 1900 and to matriculate there officially in 1904.
  • Fundamental Theorems in Mathematics

    Fundamental Theorems in Mathematics

    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
  • Optimal Control for Automotive Powertrain Applications

    Optimal Control for Automotive Powertrain Applications

    Universitat Politecnica` de Valencia` Departamento de Maquinas´ y Motores Termicos´ Optimal Control for Automotive Powertrain Applications PhD Dissertation Presented by Alberto Reig Advised by Dr. Benjam´ınPla Valencia, September 2017 PhD Dissertation Optimal Control for Automotive Powertrain Applications Presented by Alberto Reig Advised by Dr. Benjam´ınPla Examining committee President: Dr. Jos´eGalindo Lucas Secretary: Dr. Octavio Armas Vergel Vocal: Dr. Marcello Canova Valencia, September 2017 To the geniuses of the past, who built our future; to those names in the shades, bringing our knowledge. They not only supported this work but also the world we live in. If you can solve it, it is an exercise; otherwise it's a research problem. | Richard Bellman e Resumen El Control Optimo´ (CO) es esencialmente un problema matem´aticode b´us- queda de extremos, consistente en la definici´onde un criterio a minimizar (o maximizar), restricciones que deben satisfacerse y condiciones de contorno que afectan al sistema. La teor´ıade CO ofrece m´etodos para derivar una trayectoria de control que minimiza (o maximiza) ese criterio. Esta Tesis trata la aplicaci´ondel CO en automoci´on,y especialmente en el motor de combusti´oninterna. Las herramientas necesarias son un m´etodo de optimizaci´ony una representaci´onmatem´aticade la planta motriz. Para ello, se realiza un an´alisiscuantitativo de las ventajas e inconvenientes de los tres m´etodos de optimizaci´onexistentes en la literatura: programaci´ondin´amica, principio m´ınimode Pontryagin y m´etodos directos. Se desarrollan y describen los algoritmos para implementar estos m´etodos as´ıcomo un modelo de planta motriz, validado experimentalmente, que incluye la din´amicalongitudinal del veh´ıculo,modelos para el motor el´ectricoy las bater´ıas,y un modelo de motor de combusti´onde valores medios.
  • Notices of the American Mathematical Society ABCD Springer.Com

    Notices of the American Mathematical Society ABCD Springer.Com

    ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook Collection of the American Mathematical Society August 2009 Volume 56, Number 7 Guido Castelnuovo and Francesco Severi: NEW NEW NEW Two Personalities, Two The objective of this textbook is the Blackjack is among the most popular This second edition of Alexander Soifer’s Letters construction, analysis, and interpreta- casino table games, one where astute How Does One Cut a Triangle? tion of mathematical models to help us choices of playing strategy can create demonstrates how different areas of page 800 understand the world we live in. an advantage for the player. Risk and mathematics can be juxtaposed in the Students and researchers interested in Reward analyzes the game in depth, solution of a given problem. The author mathematical modelling in math- pinpointing not just its optimal employs geometry, algebra, trigono- ematics, physics, engineering and the strategies but also its financial metry, linear algebra, and rings to The Dixmier–Douady applied sciences will find this text useful. performance, in terms of both expected develop a miniature model of cash flow and associated risk. mathematical research. Invariant for Dummies 2009. Approx. 480 p. (Texts in Applied Mathematics, Vol. 56) Hardcover 2009. Approx. 140 p. 23 illus. Hardcover 2nd ed. 2009. XXX, 174 p. 80 illus. Softcover page 809 ISBN 978-0-387-87749-5 7 $69.95 ISBN 978-1-4419-0252-8 7 $49.95 ISBN 978-0-387-74650-0 7 approx. $24.95 For access check with your librarian Waco Meeting page 879 A Primer on Scientific Data Mining in Agriculture Explorations in Monte Programming with Python A.
  • Agms20170007.Pdf

    Agms20170007.Pdf

    This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Le Donne, Enrico Title: A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries Year: 2017 Version: Please cite the original version: Le Donne, E. (2017). A Primer on Carnot Groups: Homogenous Groups, Carnot- Carathéodory Spaces, and Regularity of Their Isometries. Analysis and Geometry in Metric Spaces, 5(1), 116-137. https://doi.org/10.1515/agms-2017-0007 All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Anal. Geom. Metr. Spaces 2017; 5:116–137 Survey Paper Open Access Enrico Le Donne* A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries https://doi.org/10.1515/agms-2017-0007 Received November 18, 2016; revised September 7, 2017; accepted November 8, 2017 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.
  • 1 Background and History 1.1 Classifying Exotic Spheres the Kervaire-Milnor Classification of Exotic Spheres

    1 Background and History 1.1 Classifying Exotic Spheres the Kervaire-Milnor Classification of Exotic Spheres

    A historical introduction to the Kervaire invariant problem ESHT boot camp April 4, 2016 Mike Hill University of Virginia Mike Hopkins 1.1 Harvard University Doug Ravenel University of Rochester Mike Hill, myself and Mike Hopkins Photo taken by Bill Browder February 11, 2010 1.2 1.3 1 Background and history 1.1 Classifying exotic spheres The Kervaire-Milnor classification of exotic spheres 1 About 50 years ago three papers appeared that revolutionized algebraic and differential topology. John Milnor’s On manifolds home- omorphic to the 7-sphere, 1956. He constructed the first “exotic spheres”, manifolds homeomorphic • but not diffeomorphic to the stan- dard S7. They were certain S3-bundles over S4. 1.4 The Kervaire-Milnor classification of exotic spheres (continued) • Michel Kervaire 1927-2007 Michel Kervaire’s A manifold which does not admit any differentiable structure, 1960. His manifold was 10-dimensional. I will say more about it later. 1.5 The Kervaire-Milnor classification of exotic spheres (continued) • Kervaire and Milnor’s Groups of homotopy spheres, I, 1963. They gave a complete classification of exotic spheres in dimensions ≥ 5, with two caveats: (i) Their answer was given in terms of the stable homotopy groups of spheres, which remain a mystery to this day. (ii) There was an ambiguous factor of two in dimensions congruent to 1 mod 4. The solution to that problem is the subject of this talk. 1.6 1.2 Pontryagin’s early work on homotopy groups of spheres Pontryagin’s early work on homotopy groups of spheres Back to the 1930s Lev Pontryagin 1908-1988 Pontryagin’s approach to continuous maps f : Sn+k ! Sk was • Assume f is smooth.
  • The Riemann Hypothesis in Characteristic P in Historical Perspective

    The Riemann Hypothesis in Characteristic P in Historical Perspective

    The Riemann Hypothesis in Characteristic p in Historical Perspective Peter Roquette Version of 3.7.2018 2 Preface This book is the result of many years' work. I am telling the story of the Riemann hypothesis for function fields, or curves, of characteristic p starting with Artin's thesis in the year 1921, covering Hasse's work in the 1930s on elliptic fields and more, until Weil's final proof in 1948. The main sources are letters which were exchanged among the protagonists during that time which I found in various archives, mostly in the University Library in G¨ottingen but also at other places in the world. I am trying to show how the ideas formed, and how the proper notions and proofs were found. This is a good illustration, fortunately well documented, of how mathematics develops in general. Some of the chapters have already been pre-published in the \Mitteilungen der Mathematischen Gesellschaft in Hamburg". But before including them into this book they have been thoroughly reworked, extended and polished. I have written this book for mathematicians but essentially it does not require any special knowledge of particular mathematical fields. I have tried to explain whatever is needed in the particular situation, even if this may seem to be superfluous to the specialist. Every chapter is written such that it can be read independently of the other chapters. Sometimes this entails a repetition of information which has al- ready been given in another chapter. The chapters are accompanied by \summaries". Perhaps it may be expedient first to look at these summaries in order to get an overview of what is going on.
  • Wolf Barth (1942--2016)

    Wolf Barth (1942--2016)

    Wolf Barth (1942{2016) Thomas Bauer, Klaus Hulek, Slawomir Rams, Alessandra Sarti, Tomasz Szemberg 1. Life 1.1. Youth Wolf Paul Barth was born on 20th October 1942 in Wernigerode, a small town in the Harz mountains. This is where his family, originally living in Nuremberg, had sought refuge from the bombing raids. In 1943 the family moved to Simmelsdorf, which is close to Nuremberg, but less vulnerable to attacks. In 1945 Barth's brother Hannes was born. The family, the father was a town employee, then moved back to Nuremberg and Wolf Barth's home was close to the famous Nuremberg castle { his playgrounds were the ruins of Nuremberg. In 1961 Wolf Barth successfully completed the Hans Sachs Gymnasium in Nuremberg. He was an excellent student (except in sports), and he then chose to study mathemat- ics and physics at the nearby Universit¨atErlangen, officially called Friedrich-Alexander Universit¨atErlangen-N¨urnberg. 1.2. Studies and early academic career As it was the standard at the time, Wolf Barth enroled for the Staatsexamen, the German high school teacher's examination. At this time, Reinhold Remmert was a professor of mathematics in Erlangen and Wolf Barth soon felt attracted to this area of mathematics. So, when Remmert moved to G¨ottingenin 1963, Barth went with him. G¨ottingen had at that time started to recover to some extent after its losses in the Nazi period and Hans Grauert was one on the leading figures who attracted many talented mathematicians. And indeed, it was Grauert, who became Barth's second academic teacher and a figure who influenced his mathematical thinking greatly.
  • 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.