Letters from William Burnside to Robert Fricke: Automorphic Functions, and the Emergence of the Burnside Problem
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
Permutation Groups Containing a Regular Abelian Subgroup: The
PERMUTATION GROUPS CONTAINING A REGULAR ABELIAN SUBGROUP: THE TANGLED HISTORY OF TWO MISTAKES OF BURNSIDE MARK WILDON Abstract. A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2- transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later in 1921 he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this note we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data. 1. Introduction In 1911, writing in §252 of the second edition of his influential textbook [6], Burnside claimed a proof of the following theorem. Theorem 1.1. Let G be a transitive permutation group of composite prime- power degree containing a regular cyclic subgroup. Either G is imprimitive or G is 2-transitive. An error in the penultimate sentence of Burnside’s proof was noted in [7, page 24], where Neumann remarks ‘Nevertheless, the theorem is certainly true and can be proved by similar character-theoretic methods to those that Burnside employed’. In §3 we present the correct part of Burnside’s arXiv:1705.07502v2 [math.GR] 11 Jun 2017 proof in today’s language. -
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]. -
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. -
Introduction to Representation Theory
STUDENT MATHEMATICAL LIBRARY Volume 59 Introduction to Representation Theory Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina with historical interludes by Slava Gerovitch http://dx.doi.org/10.1090/stml/059 STUDENT MATHEMATICAL LIBRARY Volume 59 Introduction to Representation Theory Pavel Etingof Oleg Golberg Sebastian Hensel Tiankai Liu Alex Schwendner Dmitry Vaintrob Elena Yudovina with historical interludes by Slava Gerovitch American Mathematical Society Providence, Rhode Island Editorial Board Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell 2010 Mathematics Subject Classification. Primary 16Gxx, 20Gxx. For additional information and updates on this book, visit www.ams.org/bookpages/stml-59 Library of Congress Cataloging-in-Publication Data Introduction to representation theory / Pavel Etingof ...[et al.] ; with historical interludes by Slava Gerovitch. p. cm. — (Student mathematical library ; v. 59) Includes bibliographical references and index. ISBN 978-0-8218-5351-1 (alk. paper) 1. Representations of algebras. 2. Representations of groups. I. Etingof, P. I. (Pavel I.), 1969– QA155.I586 2011 512.46—dc22 2011004787 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294 USA. -
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. -
The Legacy of Felix Klein
The Legacy of Thematic Afternoon Felix Klein The Legacy of Felix Klein The Legacy of The legacy of Felix Klein Felix Klein First Hour: Panel (15.00–16.00 h) 1. Introduction: What is and what could be the legacy of Felix Klein? Hans-Georg Weigand (Würzburg, GER) 2. Felix Klein - Biographical Notes Renate Tobies (Jena, GER): 3. Introduction to the Two Hour Session (16.30–18.30 h): • Strand A: Functional thinking Bill McCallum (Arizona, USA). • Strand B: Intuitive thinking and visualization Michael Neubrand (Oldenburg, GER). • Strand C: Elementary Mathematics from a higher standpoint Marta Menghini (Rome, IT) & Gert Schubring (Bielefeld, GER). 1 The Legacy of The legacy of Felix Klein Felix Klein First Hour: Panel (15.00–16.00 h) 1. Introduction: What is and what could be the legacy of Felix Klein? Hans-Georg Weigand (Würzburg, GER) 2. Felix Klein - Biographical Notes Renate Tobies (Jena, GER): 3. Introduction to the Two Hour Session (16.30–18.30 h): • Strand A: Functional thinking Bill McCallum (Arizona, USA). • Strand B: Intuitive thinking and visualization Michael Neubrand (Oldenburg, GER). • Strand C: Elementary Mathematics from a higher standpoint Marta Menghini (Rome, IT) & Gert Schubring (Bielefeld, GER). The Legacy of The legacy of Felix Klein Felix Klein First Hour: Panel (15.00–16.00 h) 1. Introduction: What is and what could be the legacy of Felix Klein? Hans-Georg Weigand (Würzburg, GER) 2. Felix Klein - Biographical Notes Renate Tobies (Jena, GER): 3. Introduction to the Two Hour Session (16.30–18.30 h): • Strand A: Functional thinking Bill McCallum (Arizona, USA). -
The Characters and Commutators of Finite Groups
THE CHARACTERS AND COMMUTATORS OF FINITE GROUPS By TIM W. BONNER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 1 °c 2009 Tim W. Bonner 2 ACKNOWLEDGMENTS I am sincerely grateful to my adviser, Alexandre Turull. With unyielding patience and constant support, he has been integral in my academic and personal growth. I feel indebted to have been his student, and I am fortunate to know such an inspiring individual. I also will forever appreciate the encouragement of my family. Over the past six years, my parents and sister have responded to each step forward and every setback with calm assurance. Finally, my wife, Emily, has witnessed it all only a glance away, and this work undeniably bears the steady strength of her hand. 3 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 3 LIST OF TABLES ..................................... 5 ABSTRACT ........................................ 6 CHAPTER 1 INTRODUCTION .................................. 8 1.1 Products of Commutators ........................... 8 1.2 Further Generation Problems ......................... 9 1.2.1 Products of Conjugacy Classes ..................... 9 1.2.2 Products of Characters ......................... 10 1.3 The Taketa Problem .............................. 11 2 MATHEMATICAL PRELIMINARIES FOR CHAPTERS 3 AND 4 ....... 13 3 PRODUCTS OF COMMUTATORS AND BARDAKOV'S CONJECTURE ... 17 3.1 A Character Identity of Burnside ....................... 18 3.2 Analysis and Results .............................. 21 3.3 Bardakov's Conjecture ............................. 26 4 PRODUCTS OF CONJUGACY CLASSES AND CHARACTERS ........ 28 4.1 Conjugacy Class Covering Numbers ...................... 28 4.2 Comparison with Previous Bounds ...................... 33 4.3 Character Covering Numbers ........................ -
Regular Permutation Groups and Cayley Graphs Cheryl E Praeger University of Western Australia
Regular permutation groups and Cayley graphs Cheryl E Praeger University of Western Australia 1 What are Cayley graphs? Group : G with generating set S = s, t, u, . 1 { } Group elements: ‘words in S’ stu− s etc vertices: group elements Cayley graph: Cay(G, S) edges: multiplication from S 2 What are Cayley graphs? If S inverse closed: s S s 1 S ∈ ⇐⇒ − ∈ Cayley graph Cay(G, S): may use undirected edges 3 Some reasonable questions Where: do they arise in mathematics today? Where: did they originate? What: kinds of groups G give interesting Cayley graphs Cay(G, S)? Which graphs: arise as Cayley graphs? Does it matter: what S we choose? Are: Cayley graphs important and why? And: what about regular permutation groups? Let’s see how I go with answers! 4 In Topology: Embedding maps in surfaces Thanks to Ethan Hein: flickr.com 5 Computer networks; experimental layouts (statistics) Thanks to Ethan Hein: flickr.com and Jason Twamley 6 Random walks on Cayley graphs Applications: from percolation theory to group computation How large n: g ‘approximately random’ in G? Independence? Method: for random selection in groups – underpins randomised algorithms for group computation (Babai, 1991) 7 Fundamental importance for group actions in combinatorics and geometry Iwilldescribe: Regular permutation groups Origins of Cayley graphs Links with group theory Some recent work and open problems on primitive Cayley graphs 8 Permutation groups Permutation : of set Ω, bijection g :Ω Ω Symmetric group of all permutations→ of Ω group Sym (Ω): under composition, for