The “Wide Influence” of Leonard Eugene Dickson
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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. -
Tōhoku Rick Jardine
INFERENCE / Vol. 1, No. 3 Tōhoku Rick Jardine he publication of Alexander Grothendieck’s learning led to great advances: the axiomatic description paper, “Sur quelques points d’algèbre homo- of homology theory, the theory of adjoint functors, and, of logique” (Some Aspects of Homological Algebra), course, the concepts introduced in Tōhoku.5 Tin the 1957 number of the Tōhoku Mathematical Journal, This great paper has elicited much by way of commen- was a turning point in homological algebra, algebraic tary, but Grothendieck’s motivations in writing it remain topology and algebraic geometry.1 The paper introduced obscure. In a letter to Serre, he wrote that he was making a ideas that are now fundamental; its language has with- systematic review of his thoughts on homological algebra.6 stood the test of time. It is still widely read today for the He did not say why, but the context suggests that he was clarity of its ideas and proofs. Mathematicians refer to it thinking about sheaf cohomology. He may have been think- simply as the Tōhoku paper. ing as he did, because he could. This is how many research One word is almost always enough—Tōhoku. projects in mathematics begin. The radical change in Gro- Grothendieck’s doctoral thesis was, by way of contrast, thendieck’s interests was best explained by Colin McLarty, on functional analysis.2 The thesis contained important who suggested that in 1953 or so, Serre inveigled Gro- results on the tensor products of topological vector spaces, thendieck into working on the Weil conjectures.7 The Weil and introduced mathematicians to the theory of nuclear conjectures were certainly well known within the Paris spaces. -
The Probl\Eme Des M\'Enages Revisited
The probl`emedes m´enages revisited Lefteris Kirousis and Georgios Kontogeorgiou Department of Mathematics National and Kapodistrian University of Athens [email protected], [email protected] August 13, 2018 Abstract We present an alternative proof to the Touchard-Kaplansky formula for the probl`emedes m´enages, which, we believe, is simpler than the extant ones and is in the spirit of the elegant original proof by Kaplansky (1943). About the latter proof, Bogart and Doyle (1986) argued that despite its cleverness, suffered from opting to give precedence to one of the genders for the couples involved (Bogart and Doyle supplied an elegant proof that avoided such gender-dependent bias). The probl`emedes m´enages (the couples problem) asks to count the number of ways that n man-woman couples can sit around a circular table so that no one sits next to her partner or someone of the same gender. The problem was first stated in an equivalent but different form by Tait [11, p. 159] in the framework of knot theory: \How many arrangements are there of n letters, when A cannot be in the first or second place, B not in the second or third, &c." It is easy to see that the m´enageproblem is equivalent to Tait's arrangement problem: first sit around the table, in 2n! ways, the men or alternatively the women, leaving an empty space between any two consecutive of them, then count the number of ways to sit the members of the other gender. Recurrence relations that compute the answer to Tait's question were soon given by Cayley and Muir [2,9,3, 10]. -
Roots of L-Functions of Characters Over Function Fields, Generic
Roots of L-functions of characters over function fields, generic linear independence and biases Corentin Perret-Gentil Abstract. We first show joint uniform distribution of values of Kloost- erman sums or Birch sums among all extensions of a finite field Fq, for ˆ almost all couples of arguments in Fq , as well as lower bounds on dif- ferences. Using similar ideas, we then study the biases in the distribu- tion of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick–Waxman and Keating–Rudnick, building on cohomological in- terpretations and determinations of monodromy groups by Katz. Our results are based on generic linear independence of Frobenius eigenval- ues of ℓ-adic representations, that we obtain from integral monodromy information via the strategy of Kowalski, which combines his large sieve for Frobenius with a method of Girstmair. An extension of the large sieve is given to handle wild ramification of sheaves on varieties. Contents 1. Introduction and statement of the results 1 2. Kloosterman sums and Birch sums 12 3. Angles of Gaussian primes 14 4. Prime polynomials in short intervals 20 5. An extension of the large sieve for Frobenius 22 6. Generic maximality of splitting fields and linear independence 33 7. Proof of the generic linear independence theorems 36 References 37 1. Introduction and statement of the results arXiv:1903.05491v2 [math.NT] 17 Dec 2019 Throughout, p will denote a prime larger than 5 and q a power of p. ˆ 1.1. Kloosterman and Birch sums. -
President's Report
Newsletter Volume 42, No. 5 • SePTemBeR–oCT oBeR 2012 PRESIDENT’S REPORT AWM Prize Winners. I am pleased to announce the winners of two of the three major prizes given by AWM at the Joint Mathematics Meetings (JMM). The Louise Hay Award is given to Amy Cohen, Rutgers University, in recognition of The purpose of the Association her “contributions to mathematics education through her writings, her talks, and for Women in Mathematics is her outstanding service to professional organizations.” The Gweneth Humphreys • to encourage women and girls to Award is given to James Morrow, University of Washington, in recognition of study and to have active careers in his “outstanding achievements in inspiring undergraduate women to discover and the mathematical sciences, and • to promote equal opportunity and pursue their passion for mathematics.” I quote from the citations written by the the equal treatment of women and selection committees and extend warm congratulations to the honorees. These girls in the mathematical sciences. awards will be presented at the Prize Session at JMM 2013 in San Diego. Congratulations to Raman Parimala, Emory University, who will present the AWM Noether Lecture at JMM 2013, entitled “A Hasse Principle for Quadratic Forms over Function Fields.” She is honored for her groundbreaking research in theory of quadratic forms, hermitian forms, linear algebraic groups, and Galois cohomology. The Noether Lecture is the oldest of the three named AWM Lectures. (The Falconer Lecture is presented at MathFest, and the Sonia Kovalevsky at the SIAM Annual Meeting.) The first Noether Lecture was given by F. Jessie MacWilliams in 1980 in recognition of her fundamental contributions to the theory of error correcting codes. -
Janko's Sporadic Simple Groups
Janko’s Sporadic Simple Groups: a bit of history Algebra, Geometry and Computation CARMA, Workshop on Mathematics and Computation Terry Gagen and Don Taylor The University of Sydney 20 June 2015 Fifty years ago: the discovery In January 1965, a surprising announcement was communicated to the international mathematical community. Zvonimir Janko, working as a Research Fellow at the Institute of Advanced Study within the Australian National University had constructed a new sporadic simple group. Before 1965 only five sporadic simple groups were known. They had been discovered almost exactly one hundred years prior (1861 and 1873) by Émile Mathieu but the proof of their simplicity was only obtained in 1900 by G. A. Miller. Finite simple groups: earliest examples É The cyclic groups Zp of prime order and the alternating groups Alt(n) of even permutations of n 5 items were the earliest simple groups to be studied (Gauss,≥ Euler, Abel, etc.) É Evariste Galois knew about PSL(2,p) and wrote about them in his letter to Chevalier in 1832 on the night before the duel. É Camille Jordan (Traité des substitutions et des équations algébriques,1870) wrote about linear groups defined over finite fields of prime order and determined their composition factors. The ‘groupes abéliens’ of Jordan are now called symplectic groups and his ‘groupes hypoabéliens’ are orthogonal groups in characteristic 2. É Émile Mathieu introduced the five groups M11, M12, M22, M23 and M24 in 1861 and 1873. The classical groups, G2 and E6 É In his PhD thesis Leonard Eugene Dickson extended Jordan’s work to linear groups over all finite fields and included the unitary groups. -
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. -
License Or Copyright Restrictions May Apply to Redistribution; See Https
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EMIL ARTIN BY RICHARD BRAUER Emil Artin died of a heart attack on December 20, 1962 at the age of 64. His unexpected death came as a tremendous shock to all who knew him. There had not been any danger signals. It was hard to realize that a person of such strong vitality was gone, that such a great mind had been extinguished by a physical failure of the body. Artin was born in Vienna on March 3,1898. He grew up in Reichen- berg, now Tschechoslovakia, then still part of the Austrian empire. His childhood seems to have been lonely. Among the happiest periods was a school year which he spent in France. What he liked best to remember was his enveloping interest in chemistry during his high school days. In his own view, his inclination towards mathematics did not show before his sixteenth year, while earlier no trace of mathe matical aptitude had been apparent.1 I have often wondered what kind of experience it must have been for a high school teacher to have a student such as Artin in his class. During the first world war, he was drafted into the Austrian Army. After the war, he studied at the University of Leipzig from which he received his Ph.D. in 1921. He became "Privatdozent" at the Univer sity of Hamburg in 1923. -
Emil Artin in America
MATHEMATICAL PERSPECTIVES BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 2, April 2013, Pages 321–330 S 0273-0979(2012)01398-8 Article electronically published on December 18, 2012 CREATING A LIFE: EMIL ARTIN IN AMERICA DELLA DUMBAUGH AND JOACHIM SCHWERMER 1. Introduction In January 1933, Adolf Hitler and the Nazi party assumed control of Germany. On 7 April of that year the Nazis created the notion of “non-Aryan descent”.1 “It was only a question of time”, Richard Brauer would later describe it, “until [Emil] Artin, with his feeling for individual freedom, his sense of justice, his abhorrence of physical violence would leave Germany” [5, p. 28]. By the time Hitler issued the edict on 26 January 1937, which removed any employee married to a Jew from their position as of 1 July 1937,2 Artin had already begun to make plans to leave Germany. Artin had married his former student, Natalie Jasny, in 1929, and, since she had at least one Jewish grandparent, the Nazis classified her as Jewish. On 1 October 1937, Artin and his family arrived in America [19, p. 80]. The surprising combination of a Roman Catholic university and a celebrated American mathematician known for his gnarly personality played a critical role in Artin’s emigration to America. Solomon Lefschetz had just served as AMS president from 1935–1936 when Artin came to his attention: “A few days ago I returned from a meeting of the American Mathematical Society where as President, I was particularly well placed to know what was going on”, Lefschetz wrote to the president of Notre Dame on 12 January 1937, exactly two weeks prior to the announcement of the Hitler edict that would influence Artin directly. -
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. -
Perspective the PNAS Way Back Then
Proc. Natl. Acad. Sci. USA Vol. 94, pp. 5983–5985, June 1997 Perspective The PNAS way back then Saunders Mac Lane, Chairman Proceedings Editorial Board, 1960–1968 Department of Mathematics, University of Chicago, Chicago, IL 60637 Contributed by Saunders Mac Lane, March 27, 1997 This essay is to describe my experience with the Proceedings of ings. Bronk knew that the Proceedings then carried lots of math. the National Academy of Sciences up until 1970. Mathemati- The Council met. Detlev was not a man to put off till to cians and other scientists who were National Academy of tomorrow what might be done today. So he looked about the Science (NAS) members encouraged younger colleagues by table in that splendid Board Room, spotted the only mathe- communicating their research announcements to the Proceed- matician there, and proposed to the Council that I be made ings. For example, I can perhaps cite my own experiences. S. Chairman of the Editorial Board. Then and now the Council Eilenberg and I benefited as follows (communicated, I think, did not often disagree with the President. Probably nobody by M. H. Stone): ‘‘Natural isomorphisms in group theory’’ then knew that I had been on the editorial board of the (1942) Proc. Natl. Acad. Sci. USA 28, 537–543; and “Relations Transactions, then the flagship journal of the American Math- between homology and homotopy groups” (1943) Proc. Natl. ematical Society. Acad. Sci. USA 29, 155–158. Soon after this, the Treasurer of the Academy, concerned The second of these papers was the first introduction of a about costs, requested the introduction of page charges for geometrical object topologists now call an “Eilenberg–Mac papers published in the Proceedings.