From Arthur Cayley Via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and Superstrings to Cantorian Space–Time
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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. -
Matrix Lie Groups
Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2). -
Lie Group and Geometry on the Lie Group SL2(R)
INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear -
Oldandnewonthe Exceptionalgroupg2 Ilka Agricola
OldandNewonthe ExceptionalGroupG2 Ilka Agricola n a talk delivered in Leipzig (Germany) on product, the Lie bracket [ , ]; as a purely algebraic June 11, 1900, Friedrich Engel gave the object it is more accessible than the original Lie first public account of his newly discovered group G. If G happens to be a group of matrices, its description of the smallest exceptional Lie Lie algebra g is easily realized by matrices too, and group G2, and he wrote in the corresponding the Lie bracket coincides with the usual commuta- Inote to the Royal Saxonian Academy of Sciences: tor of matrices. In Killing’s and Lie’s time, no clear Moreover, we hereby obtain a direct defi- distinction was made between the Lie group and nition of our 14-dimensional simple group its Lie algebra. For his classification, Killing chose [G2] which is as elegant as one can wish for. a maximal set h of linearly independent, pairwise 1 [En00, p. 73] commuting elements of g and constructed base Indeed, Engel’s definition of G2 as the isotropy vectors Xα of g (indexed over a finite subset R of group of a generic 3-form in 7 dimensions is at elements α ∈ h∗, the roots) on which all elements the basis of a rich geometry that exists only on of h act diagonally through [ , ]: 7-dimensional manifolds, whose full beauty has been unveiled in the last thirty years. [H,Xα] = α(H)Xα for all H ∈ h. This article is devoted to a detailed historical In order to avoid problems when doing so he chose and mathematical account of G ’s first years, in 2 the complex numbers C as the ground field. -
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). -
Chapter 18 Metrics, Connections, and Curvature on Lie Groups
Chapter 18 Metrics, Connections, and Curvature on Lie Groups 18.1 Left (resp. Right) Invariant Metrics Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. This chapter makes extensive use of results from a beau- tiful paper of Milnor [38]. 831 832 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS Definition 18.1. Ametric , on a Lie group G is called left-invariant (resp. right-invarianth i )i↵ u, v = (dL ) u, (dL ) v h ib h a b a b iab (resp. u, v = (dR ) u, (dR ) v ), h ib h a b a b iba for all a, b G and all u, v T G. 2 2 b ARiemannianmetricthatisbothleftandright-invariant is called a bi-invariant metric. In the sequel, the identity element of the Lie group, G, will be denoted by e or 1. 18.1. LEFT (RESP. RIGHT) INVARIANT METRICS 833 Proposition 18.1. There is a bijective correspondence between left-invariant (resp. right invariant) metrics on a Lie group G, and inner products on the Lie al- gebra g of G. If , be an inner product on g,andset h i u, v g = (dL 1)gu, (dL 1)gv , h i h g− g− i for all u, v TgG and all g G.Itisfairlyeasytocheck that the above2 induces a left-invariant2 metric on G. -
Symmetry and Mathematics
A Conceptual History of Space and Symmetry Pietro Giuseppe Fré A Conceptual History of Space and Symmetry From Plato to the Superworld 123 Pietro Giuseppe Fré University of Torino Turin, Italy ISBN 978-3-319-98022-5 ISBN 978-3-319-98023-2 (eBook) https://doi.org/10.1007/978-3-319-98023-2 Library of Congress Control Number: 2018950942 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland This book is dedicated to my family, namely to my beloved daughter Laura, to my darling wife Olga, to my young son Vladimir and to my former wife Tiziana, Laura’s mother, with whom, notwithstanding our divorce, the ties due to a common daughter and to more than twenty five years spent together remain strong. -
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. -
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). -
Full History of The
London Mathematical Society Historical Overview Taken from the Introduction to The Book of Presidents 1865-1965 ADRIAN RICE The London Mathematical Society (LMS) is the primary learned society for mathematics in Britain today. It was founded in 1865 for the promotion and extension of mathematical knowledge, and in the 140 years since its foundation this objective has remained unaltered. However, the ways in which it has been attained, and indeed the Society itself, have changed considerably during this time. In the beginning, there were just nine meetings per year, twenty-seven members and a handful of papers printed in the slim first volume of the ’s Society Proceedings. Today, with a worldwide membership in excess of two thousand, the LMS is responsible for numerous books, journals, monographs, lecture notes, and a whole range of meetings, conferences, lectures and symposia. The Society continues to uphold its original remit, but via an increasing variety of activities, ranging from advising the government on higher education, to providing financial support for a wide variety of mathematically-related pursuits. At the head of the Society there is, and always has been, a President, who is elected annually and who may serve up to two years consecutively. As befits a prestigious national organization, these Presidents have often been famous mathematicians, well known and respected by the mathematical community of their day; they include Cayley and Sylvester, Kelvin and Rayleigh, Hardy and Littlewood, Atiyah and Zeeman.1 But among the names on the presidential role of honour are many people who are perhaps not quite so famous today, ’t have theorems named after them, and who are largely forgotten by the majority of who don modern-day mathematicians. -
Arthur Cayley1
Chapter 5 ARTHUR CAYLEY1 (1821-1895) Arthur Cayley was born at Richmond in Surrey, England, on August 16, 1821. His father, Henry Cayley, was descended from an ancient Yorkshire family, but had settled in St. Petersburg, Russia, as a merchant. His mother was Maria Antonia Doughty, a daughter of William Doughty; who, according to some writers, was a Russian; but her father’s name indicates an English origin. Arthur spent the first eight years of his life in St. Petersburg. In 1829 his parents took up their permanent abode at Blackheath, near London; and Arthur was sent to a private school. He early showed great liking for, and aptitude in, numerical calculations. At the age of 14 he was sent to King’s College School, London; the master of which, having observed indications of mathematical genius, advised the father to educate his son, not for his own business, as he had at first intended, but to enter the University of Cambridge. At the unusually early age of 17 Cayley began residence at Trinity College, Cambridge. As an undergraduate he had generally the reputation of being a mere mathematician; his chief diversion was novel-reading. He was also fond of travelling and mountain climbing, and was a member of the Alpine Club. The cause of the Analytical Society had now triumphed, and the Cambridge Mathematical Journal had been instituted by Gregory and Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects which had been suggested by reading the M´ecanique analytique of Lagrange and some of the works of Laplace. -
E8, the MOST EXCEPTIONAL GROUP Contents 1. Introduction 1 2. What Is
E8, THE MOST EXCEPTIONAL GROUP SKIP GARIBALDI Abstract. The five exceptional simple Lie algebras over the complex number are included one within the other as g2 ⊂ f4 ⊂ e6 ⊂ e7 ⊂ e8. The biggest one, e8, is in many ways the most mysterious. This article surveys what is known about it including many recent results, focusing on the point of view of Lie algebras and algebraic groups over fields. Contents 1. Introduction1 2. What is E8?3 3. E8 as an automorphism group5 4. Constructing the Lie algebra via gradings7 5. E8 over the real numbers9 6. E8 over an arbitrary field 12 7. Tits' construction 13 8. Cohomological invariants; the Rost invariant 15 9. The kernel of the Rost invariant; Semenov's invariant 17 10. Witt invariants 18 11. Connection with division algebras 19 12. Other recent results on E8 20 About the author 21 Acknowledgments 21 References 22 1. Introduction The Lie algebra e8 or Lie group E8 was first sighted by a human being sometime in summer or early fall of 1887, by Wilhelm Killing as part of his program to classify the semisimple finite-dimensional Lie algebras over the complex numbers [94, pp. 162{163]. Since then, it has been a source of fascination for mathematicians and others in its role as the largest of the exceptional Lie algebras. (It appears, for example, as part of the fictional Beard-Einstein Conflation in the prize-winning novel Solar [120].) Killing's classification is now considered the core of a typical graduate course on Lie algebras, and the paper containing the key ideas, [104], has Date: Version of May 16, 2016.