The Legacy of Felix Klein
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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. -
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. -
From Arthur Cayley Via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and Superstrings to Cantorian Space–Time
Available online at www.sciencedirect.com Chaos, Solitons and Fractals 37 (2008) 1279–1288 www.elsevier.com/locate/chaos From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and superstrings to Cantorian space–time L. Marek-Crnjac Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, P.O. Box 2964, SI-1001 Ljubljana, Slovenia Abstract In this work we present a historical overview of mathematical discoveries which lead to fundamental developments in super string theory, super gravity and finally to E-infinity Cantorian space–time theory. Cantorian space–time is a hierarchical fractal-like semi manifold with formally infinity many dimensions but a finite expectation number for these dimensions. The idea of hierarchy and self-similarity in science was first entertain by Right in the 18th century, later on the idea was repeated by Swedenborg and Charlier. Interestingly, the work of Mohamed El Naschie and his two contra parts Ord and Nottale was done independently without any knowledge of the above starting from non- linear dynamics and fractals. Ó 2008 Published by Elsevier Ltd. 1. Introduction Many of the profound mathematical discovery and dare I say also inventions which were made by the mathemati- cians Arthur Cayley, Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan and Emmy Noether [1] are extremely important for high energy particles in general [2] as well as in the development of E-infinity, Cantorian space–time the- ory [3,4]. The present paper is dedicated to the historical background of this subject. 2. Arthur Cayley – beginner of the group theory in the modern way Arthur Cayley was a great British mathematician. -
LMS – EPSRC Durham Symposium
LMS – EPSRC Durham Symposium Anthony Byrne Grants and Membership Administrator 12th July 2016, Durham The work of the LMS for mathematics The charitable aims of the Society: Funding the advancement of mathematical knowledge Encouraging mathematical research and collaboration ’, George Legendre Celebrating mathematical 30 Pieces achievements Publishing and disseminating mathematical knowledge Advancing and promoting mathematics The attendees of the Young Researchers in Mathematics Conference 2015, held at Oxford Historical Moments of the London Mathematical Society 1865 Foundation of LMS at University College London George Campbell De Morgan organised the first meeting, and his father, Augustus De Morgan became the 1st President 1865 First minute book list of the 27 original members 1866 LMS moves to Old Burlington House, Piccadilly J.J. Sylvester, 2nd President of the Society. 1866 Julius Plûcker Thomas Hirst Plûcker Collection of boxwood models of quartic surfaces given to Thomas Archer Hirst, Vice- President of LMS, and donated to the Society 1870 Move to Asiatic Society, 22 Albemarle Street William Spottiswoode, President 1874 Donation of £1,000 from John William Strutt (Lord Rayleigh) Generous donation enabled the Society to publish volumes of the Proceedings of the London Mathematical Society. J.W. Strutt (Lord Rayleigh), LMS President 1876-78 1881 First women members Charlotte Angas Scott and Christine Ladd 1884 First De Morgan medal awarded to Arthur Cayley 1885 Sophie Bryant First woman to have a paper published in LMS Proceedings 1916 Return to Burlington House the home of LMS until 1998 1937 ACE ’s Automatic Turing LMS Proceedings, 1937 Computing Engine, published Alan Turing’s first paper 1950 On Computable Numbers 1947 Death of G.H. -
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]. -
Arthur Cayley English Version
ARTHUR CAYLEY (August 16, 1821 – January 26, 1895) by HEINZ KLAUS STRICK, Germany Born in Richmond (Surrey), ARTHUR CAYLEY, the son of the English merchant HENRY CAYLEY, initially grew up in St Petersburg, Russia. When the family returned to England, he attended King's College in London. He entered Trinity College, Cambridge University at 17, graduated in mathematics at the age of 21 (as senior wrangler) and won the Smith's Prize, an award for students who have shown exceptional performance throughout their studies. During his studies, he published three articles in the Cambridge Mathematical Journal. In the two years after the bachelor's degree, during which he supervised new students as a tutor, there were another 28 contributions. After passing his master's degree, he decided to become a lawyer in order to earn a better living. In the following five years he worked for a well-known London notary before he was admitted to the bar in 1849. During this time he met JAMES JOSEPH SYLVESTER, who had the same interests as him. The two became friends; their conversations were always and almost exclusively about mathematical topics. CAYLEY worked as a lawyer for 14 years – and published over 250 scientific papers during this time, before he was appointed to the SADLEIRian Chair, a chair for pure mathematics at the University of Cambridge, in 1863. Although he now had only a fraction of the income he previously had as a lawyer, he was happy with the work he was able to do until his death in 1895. In total, he published 967 articles that dealt with topics from all current research areas of mathematics, and including one textbook (on JACOBI's elliptical functions). -
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. -
The Ostwald-Gibbs Correspondence: an Interesting Component in the History of the Energy Concept
114 Bull. Hist. Chem., VOLUME 27, Number 2 (2002) THE OSTWALD-GIBBS CORRESPONDENCE: AN INTERESTING COMPONENT IN THE HISTORY OF THE ENERGY CONCEPT Carl E. Moore, Alfred von Smolinski, and Bruno Jaselskis, Loyola University, Chicago In the late 1800s and early 1900s, one of the premier In support of the energy cosmological model, in figures in German chemistry and physics was the emi- 1892 he wrote (4): nent natural philosopher and scientist Wilhelm Ostwald. Thatsächlich ist die Energie das einzige Reale in der Early in his career Ostwald developed a strong interest Welt und die Materie nicht etwa ein Träger, sondern in the concept of energy from eine Erscheinungsform both the perspective of the phi- derselben. (Actually energy is losopher and the outlook of what the unique real entity (das was later to be called the physi- einzige Reale) in the world, and cal chemist. He became a leader matter is not a transporter (Träger) [of energy] but rather in a community of scientists with a state of the former.) (5) somewhat similar leanings who called themselves energeticists This energy-mass equivalence (1) . They supported and pro- concept was later stated by moted what they hoped would Einstein and by G. N. Lewis (6, become a universally accepted 7). cosmological construct based on Ostwald’s deep and abiding the unifying concept of energy interests in energetics most (2). This energy-based paradigm likely were the motivating forces embraced thermodynamics as an that caused him forcefully and integral part of its structure. tenaciously to approach Josiah As a consequence of this Willard Gibbs (who was to be- view, for a time Ostwald did not come a prominent figure in subscribe to matter-based atom- American science) with the pro- istic models and even declared posal that Gibbs rewrite and re- that the concept of matter itself publish his masterpieces on ther- was superfluous and that the in- modynamics. -
Math Horizons Math Horizons
"Searching for the 'true tree' for twenty species is like trying to find a needle in very large haystack. Yet biologists are now routinely attempting to build trees with hundreds and sometimes thousands of species." Mathematical Aspects of the'Tree of Life' Charles Semple and Mike Steel University of Canterbury, New Zealand rees have long been used for illustrating certain phe- nomena in nature. They describe the structure of braid- Ted rivers, classify acyclic hydrocarbons in chemistry, and model the growth of cell division in physiology. In evolu- tionary biology, trees describe how species evolved from a common ancestor. Evolutionary trees date back (at least) to Charles Darwin, who made an early sketch of one in a note- book from 1837, more than 20 years before his Origin of Species. In Darwin's day, the main evidence available for reconstructing such trees, apart from some fossils, was mor- phology and physiology- the physical details of different species (does an animal have wings or not, how many petals does a plant have, and so forth). As Darwin wrote in 1872: We possess no pedigrees or amorial bearings; and we have to discover and trace the many diverging lines of descent in our natural genealogies, by characters ofany kind which have long been inherited. A phylogenetic (evolutionary) tree of 42 representative mammals The problem with morphological and fossil data is that they reconstructed using a Bayesian approach by concatenating 12 pro- are patchy, and can be misleading. Also the evolution of mor- tein coding mitochondrial genes. The main mammalian groups are phological characters is often compleX: tind difficult to model.