Abraham Adrian Albert 1905-1972 by Nathan Jacobson
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The Book of Involutions
The Book of Involutions Max-Albert Knus Alexander Sergejvich Merkurjev Herbert Markus Rost Jean-Pierre Tignol @ @ @ @ @ @ @ @ The Book of Involutions Max-Albert Knus Alexander Merkurjev Markus Rost Jean-Pierre Tignol Author address: Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich,¨ Switzerland E-mail address: [email protected] URL: http://www.math.ethz.ch/~knus/ Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA E-mail address: [email protected] URL: http://www.math.ucla.edu/~merkurev/ NWF I - Mathematik, Universitat¨ Regensburg, D-93040 Regens- burg, Germany E-mail address: [email protected] URL: http://www.physik.uni-regensburg.de/~rom03516/ Departement´ de mathematique,´ Universite´ catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] URL: http://www.math.ucl.ac.be/tignol/ Contents Pr´eface . ix Introduction . xi Conventions and Notations . xv Chapter I. Involutions and Hermitian Forms . 1 1. Central Simple Algebras . 3 x 1.A. Fundamental theorems . 3 1.B. One-sided ideals in central simple algebras . 5 1.C. Severi-Brauer varieties . 9 2. Involutions . 13 x 2.A. Involutions of the first kind . 13 2.B. Involutions of the second kind . 20 2.C. Examples . 23 2.D. Lie and Jordan structures . 27 3. Existence of Involutions . 31 x 3.A. Existence of involutions of the first kind . 32 3.B. Existence of involutions of the second kind . 36 4. Hermitian Forms . 41 x 4.A. Adjoint involutions . 42 4.B. Extension of involutions and transfer . -
Bibliography
Bibliography [1] Emil Artin. Galois Theory. Dover, second edition, 1964. [2] Michael Artin. Algebra. Prentice Hall, first edition, 1991. [3] M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra. Addison Wesley, third edition, 1969. [4] Nicolas Bourbaki. Alg`ebre, Chapitres 1-3.El´ements de Math´ematiques. Hermann, 1970. [5] Nicolas Bourbaki. Alg`ebre, Chapitre 10.El´ements de Math´ematiques. Masson, 1980. [6] Nicolas Bourbaki. Alg`ebre, Chapitres 4-7.El´ements de Math´ematiques. Masson, 1981. [7] Nicolas Bourbaki. Alg`ebre Commutative, Chapitres 8-9.El´ements de Math´ematiques. Masson, 1983. [8] Nicolas Bourbaki. Elements of Mathematics. Commutative Algebra, Chapters 1-7. Springer–Verlag, 1989. [9] Henri Cartan and Samuel Eilenberg. Homological Algebra. Princeton Math. Series, No. 19. Princeton University Press, 1956. [10] Jean Dieudonn´e. Panorama des mat´ematiques pures. Le choix bourbachique. Gauthiers-Villars, second edition, 1979. [11] David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley, second edition, 1999. [12] Albert Einstein. Zur Elektrodynamik bewegter K¨orper. Annalen der Physik, 17:891–921, 1905. [13] David Eisenbud. Commutative Algebra With A View Toward Algebraic Geometry. GTM No. 150. Springer–Verlag, first edition, 1995. [14] Jean-Pierre Escofier. Galois Theory. GTM No. 204. Springer Verlag, first edition, 2001. [15] Peter Freyd. Abelian Categories. An Introduction to the theory of functors. Harper and Row, first edition, 1964. [16] Sergei I. Gelfand and Yuri I. Manin. Homological Algebra. Springer, first edition, 1999. [17] Sergei I. Gelfand and Yuri I. Manin. Methods of Homological Algebra. Springer, second edition, 2003. [18] Roger Godement. Topologie Alg´ebrique et Th´eorie des Faisceaux. -
Publications of Members, 1930-1954
THE INSTITUTE FOR ADVANCED STUDY PUBLICATIONS OF MEMBERS 1930 • 1954 PRINCETON, NEW JERSEY . 1955 COPYRIGHT 1955, BY THE INSTITUTE FOR ADVANCED STUDY MANUFACTURED IN THE UNITED STATES OF AMERICA BY PRINCETON UNIVERSITY PRESS, PRINCETON, N.J. CONTENTS FOREWORD 3 BIBLIOGRAPHY 9 DIRECTORY OF INSTITUTE MEMBERS, 1930-1954 205 MEMBERS WITH APPOINTMENTS OF LONG TERM 265 TRUSTEES 269 buH FOREWORD FOREWORD Publication of this bibliography marks the 25th Anniversary of the foundation of the Institute for Advanced Study. The certificate of incorporation of the Institute was signed on the 20th day of May, 1930. The first academic appointments, naming Albert Einstein and Oswald Veblen as Professors at the Institute, were approved two and one- half years later, in initiation of academic work. The Institute for Advanced Study is devoted to the encouragement, support and patronage of learning—of science, in the old, broad, undifferentiated sense of the word. The Institute partakes of the character both of a university and of a research institute j but it also differs in significant ways from both. It is unlike a university, for instance, in its small size—its academic membership at any one time numbers only a little over a hundred. It is unlike a university in that it has no formal curriculum, no scheduled courses of instruction, no commitment that all branches of learning be rep- resented in its faculty and members. It is unlike a research institute in that its purposes are broader, that it supports many separate fields of study, that, with one exception, it maintains no laboratories; and above all in that it welcomes temporary members, whose intellectual development and growth are one of its principal purposes. -
Oswald Veblen
NATIONAL ACADEMY OF SCIENCES O S W A L D V E B LEN 1880—1960 A Biographical Memoir by S A U N D E R S M A C L ANE Any opinions expressed in this memoir are those of the author(s) and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 1964 NATIONAL ACADEMY OF SCIENCES WASHINGTON D.C. OSWALD VEBLEN June 24,1880—August 10, i960 BY SAUNDERS MAC LANE SWALD VEBLEN, geometer and mathematical statesman, spanned O in his career the full range of twentieth-century Mathematics in the United States; his leadership in transmitting ideas and in de- veloping young men has had a substantial effect on the present mathematical scene. At the turn of the century he studied at Chi- cago, at the period when that University was first starting the doc- toral training of young Mathematicians in this country. He then continued at Princeton University, where his own work and that of his students played a leading role in the development of an outstand- ing department of Mathematics in Fine Hall. Later, when the In- stitute for Advanced Study was founded, Veblen became one of its first professors, and had a vital part in the development of this In- stitute as a world center for mathematical research. Veblen's background was Norwegian. His grandfather, Thomas Anderson Veblen, (1818-1906) came from Odegaard, Homan Con- gregation, Vester Slidre Parish, Valdris. After work as a cabinet- maker and as a Norwegian soldier, he was anxious to come to the United States. -
Split Octonions
Split Octonions Benjamin Prather Florida State University Department of Mathematics November 15, 2017 Group Algebras Split Octonions Let R be a ring (with unity). Prather Let G be a group. Loop Algebras Octonions Denition Moufang Loops An element of group algebra R[G] is the formal sum: Split-Octonions Analysis X Malcev Algebras rngn Summary gn2G Addition is component wise (as a free module). Multiplication follows from the products in R and G, distributivity and commutativity between R and G. Note: If G is innite only nitely many ri are non-zero. Group Algebras Split Octonions Prather Loop Algebras A group algebra is itself a ring. Octonions In particular, a group under multiplication. Moufang Loops Split-Octonions Analysis A set M with a binary operation is called a magma. Malcev Algebras Summary This process can be generalized to magmas. The resulting algebra often inherit the properties of M. Divisibility is a notable exception. Magmas Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Loops Split Octonions Prather Loop Algebras In particular, loops are not associative. Octonions It is useful to dene some weaker properties. Moufang Loops power associative: the sub-algebra generated by Split-Octonions Analysis any one element is associative. Malcev Algebras diassociative: the sub-algebra generated by any Summary two elements is associative. associative: the sub-algebra generated by any three elements is associative. Loops Split Octonions Prather Loop Algebras Octonions Power associativity gives us (xx)x = x(xx). Moufang Loops This allows x n to be well dened. Split-Octonions Analysis −1 Malcev Algebras Diassociative loops have two sided inverses, x . -
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. -
January 2013 Prizes and Awards
January 2013 Prizes and Awards 4:25 P.M., Thursday, January 10, 2013 PROGRAM SUMMARY OF AWARDS OPENING REMARKS FOR AMS Eric Friedlander, President LEVI L. CONANT PRIZE: JOHN BAEZ, JOHN HUERTA American Mathematical Society E. H. MOORE RESEARCH ARTICLE PRIZE: MICHAEL LARSEN, RICHARD PINK DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY DAVID P. ROBBINS PRIZE: ALEXANDER RAZBOROV TEACHING OF MATHEMATICS RUTH LYTTLE SATTER PRIZE IN MATHEMATICS: MARYAM MIRZAKHANI Mathematical Association of America LEROY P. STEELE PRIZE FOR LIFETIME ACHIEVEMENT: YAKOV SINAI EULER BOOK PRIZE LEROY P. STEELE PRIZE FOR MATHEMATICAL EXPOSITION: JOHN GUCKENHEIMER, PHILIP HOLMES Mathematical Association of America LEROY P. STEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH: SAHARON SHELAH LEVI L. CONANT PRIZE OSWALD VEBLEN PRIZE IN GEOMETRY: IAN AGOL, DANIEL WISE American Mathematical Society DAVID P. ROBBINS PRIZE FOR AMS-SIAM American Mathematical Society NORBERT WIENER PRIZE IN APPLIED MATHEMATICS: ANDREW J. MAJDA OSWALD VEBLEN PRIZE IN GEOMETRY FOR AMS-MAA-SIAM American Mathematical Society FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY ALICE T. SCHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN AN UNDERGRADUATE STUDENT: FAN WEI Association for Women in Mathematics FOR AWM LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION: AMY COHEN Association for Women in Mathematics M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE -
Obituary Nathan Jacobson (1910–1999)
OBITUARY NATHAN JACOBSON (1910–1999) Nathan Jacobson, who died on 5 December 1999, was an outstanding algebraist, whose work on almost all aspects of algebra was of fundamental importance, and whose writings will exercise a lasting influence. He had been an honorary member of the Society since 1972. Nathan Jacobson (later known as ‘Jake’ to his friends) was born in Warsaw (in what he describes as the ‘Jewish ghetto’) on 5 October 1910 (through an error some documents have the date 8 September); he was the second son of Charles Jacobson (as he would be known later) and his wife Pauline, nee´ Rosenberg. His family emigrated to the USA during the First World War, first to Nashville, Tennessee, where his father owned a small grocery store, but they then settled in Birmingham, Alabama, where Nathan received most of his schooling. Later the family moved to Columbus, Mississippi, but the young Nathan entered the University of Alabama in 1926 and graduated in 1930. His initial aim was to follow an uncle and obtain a degree in law, but at the same time he took all the (not very numerous) mathematics courses, in which he did so well that he was offered a teaching assistantship in Bull. London Math. Soc. 33 (2001) 623–630 DOI: 10.1112/S0024609301008323 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 28 Sep 2021 at 20:31:37, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0024609301008323 624 nathan jacobson mathematics in his junior (3rd) year. -
Kailash C. Misra Rutgers University, New Brunswick
Nilos Kabasilas Heinri%h von )angenstein ;emetrios Ky2ones 4lissae"s J"2ae"s Universit> 2e /aris (1060) ,eorgios /lethon ,emistos Johannes von ,m"n2en !an"el Chrysoloras (10'5) UniversitIt *ien (1.56) ,"arino 2a ?erona $asilios $essarion ,eorg von /e"erba%h (1.5') !ystras (1.06) UniversitIt *ien (1..5) Johannes 8rgyro1o"los Johannes !Lller Regiomontan"s )"%a /a%ioli UniversitC 2i /a2ova (1...) UniversitIt )ei1Big : UniversitIt *ien (1.-6) ?ittorino 2a Aeltre !arsilio Ai%ino ;omeni%o !aria Novara 2a Aerrara Cristo3oro )an2ino UniversitC 2i /a2ova (1.16) UniversitC 2i AirenBe (1.6() UniversitC 2i AirenBe (1.'0) Theo2oros ,aBes 9gnibene (9mnibon"s )eoni%en"s) $onisoli 2a )onigo 8ngelo /oliBiano Constantino1le : UniversitC 2i !antova (1.00) UniversitC 2i !antova UniversitC 2i AirenBe (1.66) ;emetrios Chal%o%on2yles R"2ol3 8gri%ola S%i1ione Aortig"erra )eo 9"ters ,aetano 2a Thiene Sigismon2o /ol%astro Thomas C Kem1is Ja%ob ben Jehiel )oans !oses /ereB !ystras : 8%%a2emia Romana (1.-() UniversitC 2egli St"2i 2i Aerrara (1.6') UniversitC 2i AirenBe (1.90) Universit> CatholiK"e 2e )o"vain (1.'-) Jan"s )as%aris Ni%oletto ?ernia /ietro Ro%%abonella Jan Stan2on%& 8le7an2er Hegi"s Johann (Johannes Ka1nion) Re"%hlin AranNois ;"bois ,irolamo (Hieronym"s 8lean2er) 8lean2ro !aarten (!artin"s ;or1i"s) van ;or1 /elo1e !atthae"s 82rian"s Jean Taga"lt UniversitC 2i /a2ova (1.6() UniversitC 2i /a2ova UniversitC 2i /a2ova CollMge Sainte@$arbe : CollMge 2e !ontaig" (1.6.) (1.6.) UniversitIt $asel : Universit> 2e /oitiers (1.66) Universit> 2e /aris (1-16) UniversitC -
Euclidean Jordan Algebras for Optimization
Euclidean Jordan algebras for optimization Michael Orlitzky Euclidean Jordan algebras for optimization Michael Orlitzky Cover image by Vexels March 23, 2021 WARNING ¡ CUIDADO ! ACHTUNG This is a draft. It contains more or less correct proofs of statements that say more or less what they should. Beyond that, I make no promises. I’ve made many, many, many, many large-scale changes to notation without ever going back to proofread from page one. Caveat emptor. 1 Contents Contents2 Glossary5 1 Preface6 1.1 Notation notation........................... 10 I Fundamentals 11 2 Vector space structure 12 2.1 Algebraic building blocks...................... 12 2.2 Vector and Hilbert spaces...................... 17 2.3 Continuity and compactness..................... 26 2.4 Algebras................................ 29 2.5 Solutions to exercises......................... 33 3 Polynomials and power-associativity 35 3.1 Univariate polynomials........................ 35 3.2 Multivariate polynomials....................... 44 3.3 Polynomial ring embeddings..................... 56 3.4 Rational functions.......................... 58 3.5 Power-associative algebras...................... 60 3.6 Polynomial continuity........................ 63 3.7 Solutions to exercises......................... 68 4 Linear Algebra 72 4.1 Linear operators, matrix representation.............. 72 4.2 Eigenvalues and self-adjoint operators............... 76 4.3 Positive semi-definite operators................... 86 4.4 Characteristic and minimal polynomials.............. 87 4.5 Solutions -
Arxiv:1412.2365V1 [Math.AG]
POLARIZATION ALGEBRAS AND THEIR RELATIONS Ualbai Umirbaev1 Abstract. Using an approach to the Jacobian Conjecture by L.M. Dru˙zkowski and K. Rusek [12], G. Gorni and G. Zampieri [19], and A.V. Yagzhev [27], we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of elements of polynomial algebras. We show that this correspondence closely relates Albert’s problem [10, Problem 1.1] in classical ring theory and the homogeneous de- pendence problem [13, page 145, Problem 7.1.5] in affine algebraic geometry related to the Jacobian Conjecture. We demonstrate these relations in concrete examples and formulate some open questions. Mathematics Subject Classification (2010): Primary 14R15, 17A40, 17A50; Sec- ondary 14R10, 17A36. Key words: the Jacobian Conjecture, polynomial mappings, homogeneous depen- dence, Engel algebras, nilpotent and solvable algebras. 1. Introduction The main objective of this paper is to connect two groups of specialists who are working on a closely connected problems and sometimes have intersections. Joining the efforts of these groups may be fruitful in studying the Jacobian Conjecture [13]. Namely, Albert’s problem [10, Problem 1.1] in classical ring theory and the homogeneous dependence prob- lem [13, page 145, Problem 7.1.5] in affine algebraic geometry are closely connected. Ring theory specialists are focused in studying only binary algebras and have some positive results in small dimensions. Affine algebraic geometry specialists have also some positive results in small dimensions and have some negative results for m-ary algebras if m ≥ 3. To make the subject more intriguing let’s start with two examples. -
Leon W. Cohen
From the collections of the Seeley G. Mudd Manuscript Library, Princeton, NJ These documents can only be used for educational and research purposes (“Fair use”) as per U.S. Copyright law (text below). By accessing this file, all users agree that their use falls within fair use as defined by the copyright law. They further agree to request permission of the Princeton University Library (and pay any fees, if applicable) if they plan to publish, broadcast, or otherwise disseminate this material. This includes all forms of electronic distribution. Inquiries about this material can be directed to: Seeley G. Mudd Manuscript Library 65 Olden Street Princeton, NJ 08540 609-258-6345 609-258-3385 (fax) [email protected] U.S. Copyright law test The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or other reproduction is not to be “used for any purpose other than private study, scholarship or research.” If a user makes a request for, or later uses, a photocopy or other reproduction for purposes in excess of “fair use,” that user may be liable for copyright infringement. • The Princeton Mathematics Community in the 1930s Transcript Number 6 (PMC6) © The Trustees of Princeton University, 1985 LEON W. COHEN (with ALBERT TUCKER) This is an interview of Leon Warren Cohen at Princeton University on 13 April 1984. The interviewers are William Aspray and Albert Tucker.