To “Applied” Mathematics: Tracing an Important Dimension of Mathematics and Its History
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Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E
BOOK REVIEW Einstein’s Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E. Rowe Einstein’s Italian modern Italy. Nor does the author shy away from topics Mathematicians: like how Ricci developed his absolute differential calculus Ricci, Levi-Civita, and the as a generalization of E. B. Christoffel’s (1829–1900) work Birth of General Relativity on quadratic differential forms or why it served as a key By Judith R. Goodstein tool for Einstein in his efforts to generalize the special theory of relativity in order to incorporate gravitation. In This delightful little book re- like manner, she describes how Levi-Civita was able to sulted from the author’s long- give a clear geometric interpretation of curvature effects standing enchantment with Tul- in Einstein’s theory by appealing to his concept of parallel lio Levi-Civita (1873–1941), his displacement of vectors (see below). For these and other mentor Gregorio Ricci Curbastro topics, Goodstein draws on and cites a great deal of the (1853–1925), and the special AMS, 2018, 211 pp. 211 AMS, 2018, vast secondary literature produced in recent decades by the world that these and other Ital- “Einstein industry,” in particular the ongoing project that ian mathematicians occupied and helped to shape. The has produced the first 15 volumes of The Collected Papers importance of their work for Einstein’s general theory of of Albert Einstein [CPAE 1–15, 1987–2018]. relativity is one of the more celebrated topics in the history Her account proceeds in three parts spread out over of modern mathematical physics; this is told, for example, twelve chapters, the first seven of which cover episodes in [Pais 1982], the standard biography of Einstein. -
The Cambridge Mathematical Journal and Its Descendants: the Linchpin of a Research Community in the Early and Mid-Victorian Age ✩
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematica 31 (2004) 455–497 www.elsevier.com/locate/hm The Cambridge Mathematical Journal and its descendants: the linchpin of a research community in the early and mid-Victorian Age ✩ Tony Crilly ∗ Middlesex University Business School, Hendon, London NW4 4BT, UK Received 29 October 2002; revised 12 November 2003; accepted 8 March 2004 Abstract The Cambridge Mathematical Journal and its successors, the Cambridge and Dublin Mathematical Journal,and the Quarterly Journal of Pure and Applied Mathematics, were a vital link in the establishment of a research ethos in British mathematics in the period 1837–1870. From the beginning, the tension between academic objectives and economic viability shaped the often precarious existence of this line of communication between practitioners. Utilizing archival material, this paper presents episodes in the setting up and maintenance of these journals during their formative years. 2004 Elsevier Inc. All rights reserved. Résumé Dans la période 1837–1870, le Cambridge Mathematical Journal et les revues qui lui ont succédé, le Cambridge and Dublin Mathematical Journal et le Quarterly Journal of Pure and Applied Mathematics, ont joué un rôle essentiel pour promouvoir une culture de recherche dans les mathématiques britanniques. Dès le début, la tension entre les objectifs intellectuels et la rentabilité économique marqua l’existence, souvent précaire, de ce moyen de communication entre professionnels. Sur la base de documents d’archives, cet article présente les épisodes importants dans la création et l’existence de ces revues. 2004 Elsevier Inc. -
Lecture Notes for Felix Klein Lectures
Preprint typeset in JHEP style - HYPER VERSION Lecture Notes for Felix Klein Lectures Gregory W. Moore Abstract: These are lecture notes for a series of talks at the Hausdorff Mathematical Institute, Bonn, Oct. 1-11, 2012 December 5, 2012 Contents 1. Prologue 7 2. Lecture 1, Monday Oct. 1: Introduction: (2,0) Theory and Physical Mathematics 7 2.1 Quantum Field Theory 8 2.1.1 Extended QFT, defects, and bordism categories 9 2.1.2 Traditional Wilsonian Viewpoint 12 2.2 Compactification, Low Energy Limit, and Reduction 13 2.3 Relations between theories 15 3. Background material on superconformal and super-Poincar´ealgebras 18 3.1 Why study these? 18 3.2 Poincar´eand conformal symmetry 18 3.3 Super-Poincar´ealgebras 19 3.4 Superconformal algebras 20 3.5 Six-dimensional superconformal algebras 21 3.5.1 Some group theory 21 3.5.2 The (2, 0) superconformal algebra SC(M1,5|32) 23 3.6 d = 6 (2, 0) super-Poincar´e SP(M1,5|16) and the central charge extensions 24 3.7 Compactification and preserved supersymmetries 25 3.7.1 Emergent symmetries 26 3.7.2 Partial Topological Twisting 26 3.7.3 Embedded Four-dimensional = 2 algebras and defects 28 N 3.8 Unitary Representations of the (2, 0) algebra 29 3.9 List of topological twists of the (2, 0) algebra 29 4. Four-dimensional BPS States and (primitive) Wall-Crossing 30 4.1 The d = 4, = 2 super-Poincar´ealgebra SP(M1,3|8). 30 N 4.2 BPS particle representations of four-dimensional = 2 superpoincar´eal- N gebras 31 4.2.1 Particle representations 31 4.2.2 The half-hypermultiplet 33 4.2.3 Long representations 34 -
Algebraic Generality Vs Arithmetic Generality in the Controversy Between C
Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874). Frédéric Brechenmacher (1). Introduction. Throughout the whole year of 1874, Camille Jordan and Leopold Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in his 1870 Traité des substitutions et des équations algébriques a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a complete set of polynomial invariants computed from the determinant |P+sQ| as a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics (2), not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be focusing in this paper on how the 1874 controversy about Jordan’s algebraic practice of canonical reduction and Kronecker’s arithmetic practice of invariant computation sheds some light on two conflicting perspectives on polynomial generality we shall appeal to former publications which have already been dealing with some of the cultural issues highlighted by this controversy such as tacit knowledge, local ways of thinking, internal philosophies and disciplinary ideals peculiar to individuals or communities [Brechenmacher 200?a] as well as the different perceptions expressed by the two opponents of a long term history involving authors such as Joseph-Louis Lagrange, Augustin Cauchy and Charles Hermite [Brechenmacher 200?b]. -
Felix Klein and His Erlanger Programm D
WDS'07 Proceedings of Contributed Papers, Part I, 251–256, 2007. ISBN 978-80-7378-023-4 © MATFYZPRESS Felix Klein and his Erlanger Programm D. Trkovsk´a Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. The present paper is dedicated to Felix Klein (1849–1925), one of the leading German mathematicians in the second half of the 19th century. It gives a brief account of his professional life. Some of his activities connected with the reform of mathematics teaching at German schools are mentioned as well. In the following text, we describe fundamental ideas of his Erlanger Programm in more detail. References containing selected papers relevant to this theme are attached. Introduction The Erlanger Programm plays an important role in the development of mathematics in the 19th century. It is the title of Klein’s famous lecture Vergleichende Betrachtungen ¨uber neuere geometrische Forschungen [A Comparative Review of Recent Researches in Geometry] which was presented on the occasion of his admission as a professor at the Philosophical Faculty of the University of Erlangen in October 1872. In this lecture, Felix Klein elucidated the importance of the term group for the classification of various geometries and presented his unified way of looking at various geometries. Klein’s basic idea is that each geometry can be characterized by a group of transformations which preserve elementary properties of the given geometry. Professional Life of Felix Klein Felix Klein was born on April 25, 1849 in D¨usseldorf,Prussia. Having finished his study at the grammar school in D¨usseldorf,he entered the University of Bonn in 1865 to study natural sciences. -
The Liouville Equation in Atmospheric Predictability
The Liouville Equation in Atmospheric Predictability Martin Ehrendorfer Institut fur¨ Meteorologie und Geophysik, Universitat¨ Innsbruck Innrain 52, A–6020 Innsbruck, Austria [email protected] 1 Introduction and Motivation It is widely recognized that weather forecasts made with dynamical models of the atmosphere are in- herently uncertain. Such uncertainty of forecasts produced with numerical weather prediction (NWP) models arises primarily from two sources: namely, from imperfect knowledge of the initial model condi- tions and from imperfections in the model formulation itself. The recognition of the potential importance of accurate initial model conditions and an accurate model formulation dates back to times even prior to operational NWP (Bjerknes 1904; Thompson 1957). In the context of NWP, the importance of these error sources in degrading the quality of forecasts was demonstrated to arise because errors introduced in atmospheric models, are, in general, growing (Lorenz 1982; Lorenz 1963; Lorenz 1993), which at the same time implies that the predictability of the atmosphere is subject to limitations (see, Errico et al. 2002). An example of the amplification of small errors in the initial conditions, or, equivalently, the di- vergence of initially nearby trajectories is given in Fig. 1, for the system discussed by Lorenz (1984). The uncertainty introduced into forecasts through uncertain initial model conditions, and uncertainties in model formulations, has been the subject of numerous studies carried out in parallel to the continuous development of NWP models (e.g., Leith 1974; Epstein 1969; Palmer 2000). In addition to studying the intrinsic predictability of the atmosphere (e.g., Lorenz 1969a; Lorenz 1969b; Thompson 1985a; Thompson 1985b), efforts have been directed at the quantification or predic- tion of forecast uncertainty that arises due to the sources of uncertainty mentioned above (see the review papers by Ehrendorfer 1997 and Palmer 2000, and Ehrendorfer 1999). -
On an Experimentum Crucis for Optics
Research Report: On an Experimentum Crucis for Optics Ionel DINU, M.Sc. Teacher of Physics E-mail: [email protected] (Dated: November 17, 2010) Abstract Formulated almost 150 years ago, Thomas Young’s hypothesis that light might be a transverse wave has never been seriously questioned, much less subjected to experiment. In this article I report on an attempt to prove experimentally that Young’s hypothesis is untenable. Although it has certain limitations, the experiment seems to show that sound in air, a longitudinal wave, can be “polarized” by reflection just like light, and this can be used as evidence against Young’s hypothesis. Further refinements of the experimental setup may yield clearer results, making this report useful to those interested in the important issue of whether light is a transverse or a longitudinal wave. Introduction In the course of development of science, there has always been a close connection between sound and light [1]. Acoustics stimulated research and discovery in optics and vice-versa. Reflection and refraction being the first common points observed between the two, the analogy between optical and acoustical phenomena has been proven also in the case of interference, diffraction, Doppler effect, and even in their mode of production: sounds are produced by the vibration of macroscopic objects, while light is produced by vibrating molecules and atoms. The wave nature of sound implies that light is also a wave and, if matter must be present for sound to be propagated, then aether must exist in order to account for the propagation of light through spaces void of all the matter known to chemistry. -
The Tangled Tale of Phase Space David D Nolte, Purdue University
Purdue University From the SelectedWorks of David D Nolte Spring 2010 The Tangled Tale of Phase Space David D Nolte, Purdue University Available at: https://works.bepress.com/ddnolte/2/ Preview of Chapter 6: DD Nolte, Galileo Unbound (Oxford, 2018) The tangled tale of phase space David D. Nolte feature Phase space has been called one of the most powerful inventions of modern science. But its historical origins are clouded in a tangle of independent discovery and misattributions that persist today. David Nolte is a professor of physics at Purdue University in West Lafayette, Indiana. Figure 1. Phase space , a ubiquitous concept in physics, is espe- cially relevant in chaos and nonlinear dynamics. Trajectories in phase space are often plotted not in time but in space—as rst- return maps that show how trajectories intersect a region of phase space. Here, such a rst-return map is simulated by a so- called iterative Lozi mapping, (x, y) (1 + y − x/2, −x). Each color represents the multiple intersections of a single trajectory starting from dierent initial conditions. Hamiltonian Mechanics is geometry in phase space. ern physics (gure 1). The historical origins have been further —Vladimir I. Arnold (1978) obscured by overly generous attribution. In virtually every textbook on dynamics, classical or statistical, the rst refer- Listen to a gathering of scientists in a hallway or a ence to phase space is placed rmly in the hands of the French coee house, and you are certain to hear someone mention mathematician Joseph Liouville, usually with a citation of the phase space. -
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. -
The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series. -
On the Origin and Early History of Functional Analysis
U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis. -
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.