Emmy Noether and the Independent Social Democratic Party
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Mathematics People
Mathematics People symmetric spaces, L²-cohomology, arithmetic groups, and MacPherson Awarded Hopf the Langlands program. The list is by far not complete, and Prize we try only to give a representative selection of his contri- bution to mathematics. He influenced a whole generation Robert MacPherson of the Institute for Advanced Study of mathematicians by giving them new tools to attack has been chosen the first winner of the Heinz Hopf Prize difficult problems and teaching them novel geometric, given by ETH Zurich for outstanding scientific work in the topological, and algebraic ways of thinking.” field of pure mathematics. MacPherson, a leading expert Robert MacPherson was born in 1944 in Lakewood, in singularities, delivered the Heinz Hopf Lectures, titled Ohio. He received his B.A. from Swarthmore College in “How Nature Tiles Space”, in October 2009. The prize also 1966 and his Ph.D. from Harvard University in 1970. He carries a cash award of 30,000 Swiss francs, approximately taught at Brown University from 1970 to 1987 and at equal to US$30,000. the Massachusetts Institute of Technology from 1987 to The following quotation was taken from a tribute to 1994. He has been at the Institute for Advanced Study in MacPherson by Gisbert Wüstholz of ETH Zurich: “Singu- Princeton since 1994. His work has introduced radically larities can be studied in different ways using analysis, new approaches to the topology of singular spaces and or you can regard them as geometric phenomena. For the promoted investigations across a great spectrum of math- latter, their study demands a deep geometric intuition ematics. -
Albert Einstein
THE COLLECTED PAPERS OF Albert Einstein VOLUME 15 THE BERLIN YEARS: WRITINGS & CORRESPONDENCE JUNE 1925–MAY 1927 Diana Kormos Buchwald, József Illy, A. J. Kox, Dennis Lehmkuhl, Ze’ev Rosenkranz, and Jennifer Nollar James EDITORS Anthony Duncan, Marco Giovanelli, Michel Janssen, Daniel J. Kennefick, and Issachar Unna ASSOCIATE & CONTRIBUTING EDITORS Emily de Araújo, Rudy Hirschmann, Nurit Lifshitz, and Barbara Wolff ASSISTANT EDITORS Princeton University Press 2018 Copyright © 2018 by The Hebrew University of Jerusalem 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 press.princeton.edu All Rights Reserved LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA (Revised for volume 15) Einstein, Albert, 1879–1955. The collected papers of Albert Einstein. German, English, and French. Includes bibliographies and indexes. Contents: v. 1. The early years, 1879–1902 / John Stachel, editor — v. 2. The Swiss years, writings, 1900–1909 — — v. 15. The Berlin years, writings and correspondence, June 1925–May 1927 / Diana Kormos Buchwald... [et al.], editors. QC16.E5A2 1987 530 86-43132 ISBN 0-691-08407-6 (v.1) ISBN 978-0-691-17881-3 (v. 15) This book has been composed in Times. The publisher would like to acknowledge the editors of this volume for providing the camera-ready copy from which this book was printed. Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources. Printed in the United States of America 13579108642 INTRODUCTION TO VOLUME 15 The present volume covers a thrilling two-year period in twentieth-century physics, for during this time matrix mechanics—developed by Werner Heisenberg, Max Born, and Pascual Jordan—and wave mechanics, developed by Erwin Schrödinger, supplanted the earlier quantum theory. -
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. -
Intellektuelle Anschauung« Figurationen Von Evidenz Zwischen Kunst Und Wissen
»Intellektuelle Anschauung« Figurationen von Evidenz zwischen Kunst und Wissen 2006-02-17 12-26-43 --- Projekt: T354.kumedi.schäfer-peters.intellektuelle ansch / Dokument: FAX ID 0242108171178570|(S. 1 ) T00_01 schmutztitel.p 10817117 2006-02-17 12-26-44 --- Projekt: T354.kumedi.schäfer-peters.intellektuelle ansch / Dokument: FAX ID 0242108171178570|(S. 2 ) T00_02 vak.p 108171178642 Sibylle Peters, Martin Jörg Schäfer (Hg.) »Intellektuelle Anschauung« Figurationen von Evidenz zwischen Kunst und Wissen 2006-02-17 12-26-45 --- Projekt: T354.kumedi.schäfer-peters.intellektuelle ansch / Dokument: FAX ID 0242108171178570|(S. 3 ) T00_03 innentitel.p 1081711786 Gedruckt mit freundlicher Unterstützung der Hamburgischen Wissenschaftlichen Stiftung und der Alexander von Humboldt-Stiftung Bibliografische Information der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.ddb.de abrufbar. © 2006 transcript Verlag, Bielefeld This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. Umschlaggestaltung und Innenlayout: Kordula Röckenhaus, Bielefeld Umschlagabbildung: »Antekythera Mechanism«, aus: The Technology Museum of Thessaloniki (2000): Astronomical Measurement Instruments from Ancient Greek Tradition, S. 41 Projektmanagement: Andreas Hüllinghorst, Bielefeld Druck: Majuskel Medienproduktion GmbH, Wetzlar ISBN 3-89942-354-2 Gedruckt auf alterungsbeständigem Papier mit chlorfrei gebleichtem Zellstoff. Besuchen Sie uns im Internet: http://www.transcript-verlag.de Bitte fordern Sie unser Gesamtverzeichnis und andere Broschüren an unter: [email protected] 2006-02-17 12-26-45 --- Projekt: T354.kumedi.schäfer-peters.intellektuelle ansch / Dokument: FAX ID 0242108171178570|(S. 4 ) T00_04 impressum.p 10817117873 Inhalt Sibylle Peters, Martin Jörg Schäfer Intellektuelle Anschauung – unmögliche Evidenz 9 Rhetorik der Anschauung Rüdiger Campe Epoche der Evidenz. -
Witt's Cancellation Theorem Seen As a Cancellation 1
WITT'S CANCELLATION THEOREM SEEN AS A CANCELLATION SUNIL K. CHEBOLU, DAN MCQUILLAN, AND JAN´ MINA´ Cˇ Dedicated to the late Professor Amit Roy Abstract. Happy birthday to the Witt ring! The year 2017 marks the 80th anniversary of Witt's famous paper containing some key results, including the Witt cancellation theorem, which form the foundation for the algebraic theory of quadratic forms. We pay homage to this paper by presenting a transparent, algebraic proof of the Witt cancellation theorem, which itself is based on a cancellation. We also present an overview of some recent spectacular work which is still building on Witt's original creation of the algebraic theory of quadratic forms. 1. Introduction The algebraic theory of quadratic forms will soon celebrate its 80th birthday. Indeed, it was 1937 when Witt's pioneering paper [24] { a mere 14 pages { first introduced many beautiful results that we love so much today. These results form the foundation for the algebraic theory of quadratic forms. In particular they describe the construction of the Witt ring itself. Thus the cute little baby, \The algebraic theory of quadratic forms" was healthy and screaming with joy, making his father Ernst Witt very proud. Grandma Emmy Noether, Figure 1. Shortly after the birth of the Algebraic Theory of Quadratic Forms. had she still been alive, would have been so delighted to see this little tyke! 1 Almost right 1Emmy Noether's male Ph.D students, including Ernst Witt, were often referred to as \Noether's boys." The reader is encouraged to consult [6] to read more about her profound influence on the development of mathematics. -
Differential Geometry in the Large Seminar Lectures New York University 1946 and Stanford University 1956 Series: Lecture Notes in Mathematics
springer.com Heinz Hopf Differential Geometry in the Large Seminar Lectures New York University 1946 and Stanford University 1956 Series: Lecture Notes in Mathematics These notes consist of two parts: Selected in York 1) Geometry, New 1946, Topics University Notes Peter Lax. by Differential in the 2) Lectures on Stanford Geometry Large, 1956, Notes J. W. University by Gray. are here with no essential They reproduced change. Heinz was a mathematician who mathema- Hopf recognized important tical ideas and new mathematical cases. In the phenomena through special the central idea the of a or difficulty problem simplest background is becomes clear. in this fashion a crystal Doing geometry usually lead serious allows this to to - joy. Hopf's great insight approach for most of the in these notes have become the st- thematics, topics I will to mention a of further try ting-points important developments. few. It is clear from these notes that laid the on Hopf emphasis po- differential 2nd ed. 1989, VIII, 192 p. Most of the results in smooth differ- hedral geometry. whose is both t1al have understanding geometry polyhedral counterparts, works I wish to mention and recent important challenging. Printed book Among those of Robert on which is much in the Connelly rigidity, very spirit R. and in - of these notes (cf. Connelly, Conjectures questions open International of Mathematicians, H- of Softcover gidity, Proceedings Congress sinki vol. 1, 407-414) 1978, . 29,95 € | £26.99 | $49.95 [1]32,05 € (D) | 32,95 € (A) | CHF 43,04 eBook 24,60 € | £20.99 | $34.99 [2]24,60 € (D) | 24,60 € (A) | CHF 34,00 Available from your library or springer.com/shop MyCopy [3] Printed eBook for just € | $ 24.99 springer.com/mycopy Error[en_EN | Export.Bookseller. -
Quadratic Forms and Their Applications
Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999. -
Jt3(S\ H. Hopf, W.K. Clifford, F. Klein
CHAPTER 20 jt3(S\ H. Hopf, W.K. Clifford, F. Klein H. Samelson* Department of Mathematics, Stanford University, Stanford, CA 94305, USA E-mail: [email protected] In 1931 there appeared the seminal paper [2] by Heinz Hopf, in which he showed that 7t2>{S^) (the third homotopy group of the two-sphere S^) is nontrivial, or more specifically that it contains an element of order oo. (His language was different. Homotopy groups had not been defined yet; E. Czech introduced them at the 1932 Congress in Zurich. Interest ingly enough, both Paul Alexandroff and Hopf persuaded him not to continue with these groups. They had different reasons for considering them as not fruitful; the one because they are Abehan, and the other (if I remember right) because there is no mechanism, like chains say, to compute them. It was not until 1936 that W. Hurewicz rediscovered them and made them respectable by proving substantial theorems with and about them.) There are two parts to the paper: The first one is the definition of what now is called the Hopf invariant and the proof of its homotopy invariance. The second consists in the presentation of an example of a map from S^ to S^ that has Hopf invariant 1 and thus represents an element of infinite order of 713(5^); it is what is now called the Hopf fibration; the inverse images of the points of S^ are great circles of S^. Taking S^ as the unit-sphere ki 1^ + k2p = 1 in C^, these circles are the intersections of S^ with the various complex lines through the origin. -
A Treatise on Quantum Clifford Algebras Contents
A Treatise on Quantum Clifford Algebras Habilitationsschrift Dr. Bertfried Fauser arXiv:math/0202059v1 [math.QA] 7 Feb 2002 Universitat¨ Konstanz Fachbereich Physik Fach M 678 78457 Konstanz January 25, 2002 To Dorothea Ida and Rudolf Eugen Fauser BERTFRIED FAUSER —UNIVERSITY OF KONSTANZ I ABSTRACT: Quantum Clifford Algebras (QCA), i.e. Clifford Hopf gebras based on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five al- ternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of Graßmann-Cayley algebras including co-meet and co-join for Graßmann-Cayley co-gebras which are very efficient and may be used in Robotics, left and right contractions, left and right co-contractions, Clifford and co-Clifford products, etc. The Chevalley deformation, using a Clif- ford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a bi-convolution. Antipode and crossing are consequences of the product and co-product structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the def- inition of non-local products and interacting Hopf gebras which are generically non-perturbative. A ‘spinorial’ generalization of the antipode is given. The non- existence of non-trivial integrals in low-dimensional Clifford co-gebras is shown. Generalized cliffordization is discussed which is based on non-exponentially gen- erated bilinear forms in general resulting in non unital, non-associative products. Reasonable assumptions lead to bilinear forms based on 2-cocycles. Cliffordiza- tion is used to derive time- and normal-ordered generating functionals for the Schwinger-Dyson hierarchies of non-linear spinor field theory and spinor electro- dynamics. -
Contributions to the History of Number Theory in the 20Th Century Author
Peter Roquette, Oberwolfach, March 2006 Peter Roquette Contributions to the History of Number Theory in the 20th Century Author: Peter Roquette Ruprecht-Karls-Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg Germany E-mail: [email protected] 2010 Mathematics Subject Classification (primary; secondary): 01-02, 03-03, 11-03, 12-03 , 16-03, 20-03; 01A60, 01A70, 01A75, 11E04, 11E88, 11R18 11R37, 11U10 ISBN 978-3-03719-113-2 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2013 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 To my friend Günther Frei who introduced me to and kindled my interest in the history of number theory Preface This volume contains my articles on the history of number theory except those which are already included in my “Collected Papers”. -
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. -
EMMY NOETHER and TOPOLOGY F. Hirzebruch1 My Knowledge
EMMY NOETHER AND TOPOLOGY F. Hirzebruch1 My knowledge about the role of Emmy Noether in Algebraic Topology stems from my encounters with Paul Alexandroff and Heinz Hopf. I studied in Z¨urich at the Eidgen¨ossische Technische Hochschule for three terms, from the summer term 1949 to the summer term 1950. Heinz Hopf and Beno Eckmann were my teachers. For the first time I attended courses on Algebraic Topology and its applications in other fields. I also learnt some facts about the famous book Topologie I by Alexandroff and Hopf [A-H] and how the cooperation of Alexandroff and Hopf started. In May 1925 Hopf received his doctor’s degree at the University of Berlin un- der Erhard Schmidt. He spent the Academic Year 1925/26 at the University of G¨ottingen where Alexandroff and Hopf met for the first time in the summer term 1926. I quote from Alexandroff [A]: ”Die Bekanntschaft zwischen Hopf und mir wurde im selben Sommer zu einer en- gen Freundschaft. Wir geh¨orten beide zum Mathematiker-Kreis um Courant und Emmy Noether, zu dieser unvergeßlichen menschlichen Gemeinschaft mit ihren Musikabenden und ihren Bootsfahrten bei und mit Courant, mit ihren ”algebraisch- topologischen” Spazierg¨angen unter der F¨uhrung von Emmy Noether und nicht zuletzt mit ihren verschiedenen Badepartien und Badeunterhaltungen, die sich in der Universit¨ats-Badeanstalt an der Leine abspielten. ... Die Kliesche Schwimmanstalt war nicht nur ein Studentenbad, sie wurde auch von vielen Universit¨atsdozenten besucht, darunter von Hilbert, Courant, Emmy Noether, Prandtl, Friedrichs, Deuring, Hans Lewy, Neugebauer und vielen anderen. Von ausw¨artigen Mathematikern seien etwa Jakob Nielsen, Harald Bohr, van der Waerden, von Neumann, Andr´e Weil als Klies st¨andige Badeg¨aste erw¨ahnt.” Here one sees G¨ottingen as a mathematical world center.