On Franco–German Relations in Mathematics, 1870–1920
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
An Historical Review of the Theoretical Development of Rigid Body Displacements from Rodrigues Parameters to the finite Twist
Mechanism and Machine Theory Mechanism and Machine Theory 41 (2006) 41–52 www.elsevier.com/locate/mechmt An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist Jian S. Dai * Department of Mechanical Engineering, School of Physical Sciences and Engineering, King’s College London, University of London, Strand, London WC2R2LS, UK Received 5 November 2004; received in revised form 30 March 2005; accepted 28 April 2005 Available online 1 July 2005 Abstract The development of the finite twist or the finite screw displacement has attracted much attention in the field of theoretical kinematics and the proposed q-pitch with the tangent of half the rotation angle has dem- onstrated an elegant use in the study of rigid body displacements. This development can be dated back to RodriguesÕ formulae derived in 1840 with Rodrigues parameters resulting from the tangent of half the rota- tion angle being integrated with the components of the rotation axis. This paper traces the work back to the time when Rodrigues parameters were discovered and follows the theoretical development of rigid body displacements from the early 19th century to the late 20th century. The paper reviews the work from Chasles motion to CayleyÕs formula and then to HamiltonÕs quaternions and Rodrigues parameterization and relates the work to Clifford biquaternions and to StudyÕs dual angle proposed in the late 19th century. The review of the work from these mathematicians concentrates on the description and the representation of the displacement and transformation of a rigid body, and on the mathematical formulation and its progress. -
Geometric Algebras for Euclidean Geometry
Geometric algebras for euclidean geometry Charles Gunn Keywords. metric geometry, euclidean geometry, Cayley-Klein construc- tion, dual exterior algebra, projective geometry, degenerate metric, pro- jective geometric algebra, conformal geometric algebra, duality, homo- geneous model, biquaternions, dual quaternions, kinematics, rigid body motion. Abstract. The discussion of how to apply geometric algebra to euclidean n-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of cru- cial but largely forgotten themes from 19th century mathematics. We ∗ then introduce the dual projectivized Clifford algebra P(Rn;0;1) (eu- clidean PGA) as the most promising homogeneous (1-up) candidate for euclidean geometry. We compare euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that euclidean PGA is the smallest structure-preserving euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include euclidean sphere primitives. We arXiv:1411.6502v4 [math.GM] 23 May 2016 conclude that euclidean PGA provides a natural transition, both scien- tifically and pedagogically, between vector space models and the more complex and powerful CGA. 1. Introduction Although noneuclidean geometry of various sorts plays a fundamental role in theoretical physics and cosmology, the overwhelming volume of practical science and engineering takes place within classical euclidean space En. For this reason it is of no small interest to establish the best computational model for this space. -
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. -
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 -
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. -
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 “Geometrical Equations:” Forgotten Premises of Felix Klein's
The “Geometrical Equations:” Forgotten Premises of Felix Klein’s Erlanger Programm François Lê October 2013 Abstract Felix Klein’s Erlanger Programm (1872) has been extensively studied by historians. If the early geometrical works in Klein’s career are now well-known, his links to the theory of algebraic equations before 1872 remain only evoked in the historiography. The aim of this paper is precisely to study this algebraic background, centered around particular equations arising from geometry, and participating on the elaboration of the Erlanger Programm. An- other result of the investigation is to complete the historiography of the theory of general algebraic equations, in which those “geometrical equations” do not appear. 1 Introduction Felix Klein’s Erlanger Programm1 is one of those famous mathematical works that are often invoked by current mathematicians, most of the time in a condensed and modernized form: doing geometry comes down to studying (transformation) group actions on various sets, while taking particular care of invariant objects.2 As rough and anachronistic this description may be, it does represent Klein’s motto to praise the importance of transformation groups and their invariants to unify and classify geometries. The Erlanger Programm has been abundantly studied by historians.3 In particular, [Hawkins 1984] questioned the commonly believed revolutionary character of the Programm, giving evi- dence of both a relative ignorance of it by mathematicians from its date of publication (1872) to 1890 and the importance of Sophus Lie’s ideas4 about continuous transformations for its elabora- tion. These ideas are only a part of the many facets of the Programm’s geometrical background of which other parts (non-Euclidean geometries, line geometry) have been studied in detail in [Rowe 1989b]. -
Letters from William Burnside to Robert Fricke: Automorphic Functions, and the Emergence of the Burnside Problem
LETTERS FROM WILLIAM BURNSIDE TO ROBERT FRICKE: AUTOMORPHIC FUNCTIONS, AND THE EMERGENCE OF THE BURNSIDE PROBLEM CLEMENS ADELMANN AND EBERHARD H.-A. GERBRACHT Abstract. Two letters from William Burnside have recently been found in the Nachlass of Robert Fricke that contain instances of Burnside's Problem prior to its first publication. We present these letters as a whole to the public for the first time. We draw a picture of these two mathematicians and describe their activities leading to their correspondence. We thus gain an insight into their respective motivations, reactions, and attitudes, which may sharpen the current understanding of professional and social interactions of the mathemat- ical community at the turn of the 20th century. 1. The simple group of order 504 { a first meeting of minds Until 1902, when the publication list of the then fifty-year-old William Burnside already encompassed 90 papers, only one of these had appeared in a non-British journal: a three-page article [5] entitled \Note on the simple group of order 504" in volume 52 of Mathematische Annalen in 1898. In this paper Burnside deduced a presentation of that group in terms of generators and relations,1 which is based on the fact that it is isomorphic to the group of linear fractional transformations with coefficients in the finite field with 8 elements. The proof presented is very concise and terse, consisting only of a succession of algebraic identities, while calculations in the concrete group of transformations are omitted. In the very same volume of Mathematische Annalen, only one issue later, there can be found a paper [18] by Robert Fricke entitled \Ueber eine einfache Gruppe von 504 Operationen" (On a simple group of 504 operations), which is on exactly the same subject. -
The Felix Klein Protocols
THE FELIX KLEIN PROTOCOLS EUGENE CHISLENKO AND YURI TSCHINKEL 1 2 EUGENE CHISLENKO AND YURI TSCHINKEL 1. The Gottingen¨ Archive Two plain shelves in G¨ottingen,in the entrance room of the Mathe- matisches Institut library, hold an astounding collection of mathemat- ical manuscripts and rare books. In this locked Giftschrank, or poi- son cabinet, stand several hundred volumes, largely handwritten and mostly unique, that form an extensive archive documenting the de- velopment of one of the world's most important mathematical centers { the home of Gauss, Riemann, Dirichlet, Klein, Hilbert, Minkowski, Courant, Weyl, and other leading mathematicians and physicists of the 19th and early 20th centuries. A recent Report on the G¨ottingen Mathematical Institute Archive [2] cites \a range of material unrivalled in quantity and quality": \No single archive is even remotely compa- rable", not only because G¨ottingenwas \the leading place for mathe- matics in the world", but also because \no other community has left such a detailed record of its activity ... usually we are lucky to have lecture lists, with no indication of the contents." The collection runs from early handwritten lectures by Dirichlet, Rie- mann and Clebsch through almost 100 volumes by Hilbert to volumes of Minkowski, Hasse and Siegel on number theory, Noether on algebra, and Max Born on quantum mechanics. But the largest and most im- pressive of its centerpieces is the Seminar-Protokolle of Felix Klein: a detailed handwritten registry, spanning over 8000 pages in 29 volumes, of 40 years of seminar lectures by him, his colleagues and students, and distinguished visitors. -
Intro to Line Geom and Kinematics
TEUBNER’S MATHEMATICAL GUIDES VOLUME 41 INTRODUCTION TO LINE GEOMETRY AND KINEMATICS BY ERNST AUGUST WEISS ASSOC. PROFESSOR AT THE RHENISH FRIEDRICH-WILHELM-UNIVERSITY IN BONN Translated by D. H. Delphenich 1935 LEIPZIG AND BERLIN PUBLISHED AND PRINTED BY B. G. TEUBNER Foreword According to Felix Klein , line geometry is the geometry of a quadratic manifold in a five-dimensional space. According to Eduard Study , kinematics – viz., the geometry whose spatial element is a motion – is the geometry of a quadratic manifold in a seven- dimensional space, and as such, a natural generalization of line geometry. The geometry of multidimensional spaces is then connected most closely with the geometry of three- dimensional spaces in two different ways. The present guide gives an introduction to line geometry and kinematics on the basis of that coupling. 2 In the treatment of linear complexes in R3, the line continuum is mapped to an M 4 in R5. In that subject, the linear manifolds of complexes are examined, along with the loci of points and planes that are linked to them that lead to their analytic representation, with the help of Weitzenböck’s complex symbolism. One application of the map gives Lie ’s line-sphere transformation. Metric (Euclidian and non-Euclidian) line geometry will be treated, up to the axis surfaces that will appear once more in ray geometry as chains. The conversion principle of ray geometry admits the derivation of a parametric representation of motions from Euler ’s rotation formulas, and thus exhibits the connection between line geometry and kinematics. The main theorem on motions and transfers will be derived by means of the elegant algebra of biquaternions. -
A Survey of the Development of Geometry up to 1870
A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics. -
LONG-TERM HISTORY and EPHEMERAL CONFIGURATIONS Catherine Goldstein
LONG-TERM HISTORY AND EPHEMERAL CONFIGURATIONS Catherine Goldstein To cite this version: Catherine Goldstein. LONG-TERM HISTORY AND EPHEMERAL CONFIGURATIONS. Interna- tional Congress of Mathematicians, Aug 2018, Rio de Janeiro, Brazil. pp.487-522. hal-02334505 HAL Id: hal-02334505 https://hal.archives-ouvertes.fr/hal-02334505 Submitted on 29 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LONG-TERM HISTORY AND EPHEMERAL CONFIGURATIONS CATHERINE GOLDSTEIN Abstract. Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathe- matics has tended to focus on local topics, over a short term-scale, and on the study of ephemeral configurations of mathematicians, theorems or practices. The first part of the paper explains why this change has taken place: a renewed interest in the connections between mathematics and society, an increased at- tention to the variety of components and aspects of mathematical work, and a critical outlook on historiography itself. The problems of a long-term history are illustrated and tested using a number of episodes in the nineteenth-century history of Hermitian forms, and finally, some open questions are proposed.