MAA's 75Th Anniversary
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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. -
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. -
A History of Mathematics in America Before 1900.Pdf
THE BOOK WAS DRENCHED 00 S< OU_1 60514 > CD CO THE CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Publication Committee GILBERT AMES BLISS DAVID RAYMOND CURTISS AUBREY JOHN KEMPNER HERBERT ELLSWORTH SLAUGHT CARUS MATHEMATICAL MONOGRAPHS are an expression of THEthe desire of Mrs. Mary Hegeler Carus, and of her son, Dr. Edward H. Carus, to contribute to the dissemination of mathe- matical knowledge by making accessible at nominal cost a series of expository presenta- tions of the best thoughts and keenest re- searches in pure and applied mathematics. The publication of these monographs was made possible by a notable gift to the Mathematical Association of America by Mrs. Carus as sole trustee of the Edward C. Hegeler Trust Fund. The expositions of mathematical subjects which the monographs will contain are to be set forth in a manner comprehensible not only to teach- ers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide circle of thoughtful people who, having a moderate acquaintance with elementary mathematics, wish to extend their knowledge without prolonged and critical study of the mathematical journals and trea- tises. The scope of this series includes also historical and biographical monographs. The Carus Mathematical Monographs NUMBER FIVE A HISTORY OF MATHEMATICS IN AMERICA BEFORE 1900 By DAVID EUGENE SMITH Professor Emeritus of Mathematics Teacliers College, Columbia University and JEKUTHIEL GINSBURG Professor of Mathematics in Yeshiva College New York and Editor of "Scripta Mathematica" Published by THE MATHEMATICAL ASSOCIATION OF AMERICA with the cooperation of THE OPEN COURT PUBLISHING COMPANY CHICAGO, ILLINOIS THE OPEN COURT COMPANY Copyright 1934 by THE MATHEMATICAL ASSOCIATION OF AMKRICA Published March, 1934 Composed, Printed and Bound by tClfe QlolUgUt* $Jrr George Banta Publishing Company Menasha, Wisconsin, U. -
Blaschke, Osgood, Wiener, Hadamard and the Early Development Of
Blaschke, Osgood, Wiener, Hadamard and the Early Development of Modern Mathematics in China Chuanming Zong Abstract. In ancient times, China made great contributions to world civi- lization and in particular to mathematics. However, modern sciences includ- ing mathematics came to China rather too late. The first Chinese university was founded in 1895. The first mathematics department in China was for- mally opened at the university only in 1913. At the beginning of the twenti- eth century, some Chinese went to Europe, the United States of America and Japan for higher education in modern mathematics and returned to China as the pioneer generation. They created mathematics departments at the Chinese universities and sowed the seeds of modern mathematics in China. In 1930s, when a dozen of Chinese universities already had mathematics departments, several leading mathematicians from Europe and USA visited China, including Wilhelm Blaschke, George D. Birkhoff, William F. Osgood, Norbert Wiener and Jacques Hadamard. Their visits not only had profound impact on the mathematical development in China, but also became social events sometimes. This paper tells the history of their visits. 2020 Mathematics Subject Classification: 01A25, 01A60. 1. Introduction and Background In 1895, Peiyang University was founded in Tianjin as a result of the westernization movement. In 1898, Peking University was founded in Beijing. They are the earliest universities in China. On May 28, 1900, Britain, USA, France, Germany, Russia, Japan, Italy and Austria invaded China to suppress the Boxer Rebellion1 which was an organized movement against foreign missionaries and their followers. On August 14, 1900, more than 20,000 soldiers of the allied force occupied the Chinese capital Beijing and the emperor’s family with some high ranking officials and servants escaped to Xi’an. -
RM Calendar 2017
Rudi Mathematici x3 – 6’135x2 + 12’545’291 x – 8’550’637’845 = 0 www.rudimathematici.com 1 S (1803) Guglielmo Libri Carucci dalla Sommaja RM132 (1878) Agner Krarup Erlang Rudi Mathematici (1894) Satyendranath Bose RM168 (1912) Boris Gnedenko 1 2 M (1822) Rudolf Julius Emmanuel Clausius (1905) Lev Genrichovich Shnirelman (1938) Anatoly Samoilenko 3 T (1917) Yuri Alexeievich Mitropolsky January 4 W (1643) Isaac Newton RM071 5 T (1723) Nicole-Reine Etable de Labrière Lepaute (1838) Marie Ennemond Camille Jordan Putnam 2002, A1 (1871) Federigo Enriques RM084 Let k be a fixed positive integer. The n-th derivative of (1871) Gino Fano k k n+1 1/( x −1) has the form P n(x)/(x −1) where P n(x) is a 6 F (1807) Jozeph Mitza Petzval polynomial. Find P n(1). (1841) Rudolf Sturm 7 S (1871) Felix Edouard Justin Emile Borel A college football coach walked into the locker room (1907) Raymond Edward Alan Christopher Paley before a big game, looked at his star quarterback, and 8 S (1888) Richard Courant RM156 said, “You’re academically ineligible because you failed (1924) Paul Moritz Cohn your math mid-term. But we really need you today. I (1942) Stephen William Hawking talked to your math professor, and he said that if you 2 9 M (1864) Vladimir Adreievich Steklov can answer just one question correctly, then you can (1915) Mollie Orshansky play today. So, pay attention. I really need you to 10 T (1875) Issai Schur concentrate on the question I’m about to ask you.” (1905) Ruth Moufang “Okay, coach,” the player agreed. -
George David Birkhoff and His Mathematical Work
GEORGE DAVID BIRKHOFF AND HIS MATHEMATICAL WORK MARSTON MORSE The writer first saw Birkhoff in the fall of 1914. The graduate stu dents were meeting the professors of mathematics of Harvard in Sever 20. Maxime Bôcher, with his square beard and squarer shoes, was presiding. In the back of the room, with a different beard but equal dignity, William Fogg Osgood was counseling a student. Dun ham Jackson, Gabriel Green, Julian Coolidge and Charles Bouton were in the business of being helpful. The thirty-year-old Birkhoff was in the front row. He seemed tall even when seated, and a friendly smile disarmed a determined face. I had no reason to speak to him, but the impression he made upon me could not be easily forgotten. His change from Princeton University to Harvard in 1912 was de cisive. Although he later had magnificent opportunities to serve as a research professor in institutions other than Harvard he elected to remain in Cambridge for life. He had been an instructor at Wisconsin from 1907 to 1909 and had profited from his contacts with Van Vleck. As a graduate student in Chicago he had known Veblen and he con tinued this friendsip in the halls of Princeton. Starting college in 1902 at the University of Chicago, he changed to Harvard, remained long enough to get an A.B. degree, and then hurried back to Chicago, where he finished his graduate work in 1907. It was in 1908 that he married Margaret Elizabeth Grafius. It was clear that Birkhoff depended from the beginning to the end on her deep understanding and encouragement. -
Council Congratulates Exxon Education Foundation
from.qxp 4/27/98 3:17 PM Page 1315 From the AMS ics. The Exxon Education Foundation funds programs in mathematics education, elementary and secondary school improvement, undergraduate general education, and un- dergraduate developmental education. —Timothy Goggins, AMS Development Officer AMS Task Force Receives Two Grants The AMS recently received two new grants in support of its Task Force on Excellence in Mathematical Scholarship. The Task Force is carrying out a program of focus groups, site visits, and information gathering aimed at developing (left to right) Edward Ahnert, president of the Exxon ways for mathematical sciences departments in doctoral Education Foundation, AMS President Cathleen institutions to work more effectively. With an initial grant Morawetz, and Robert Witte, senior program officer for of $50,000 from the Exxon Education Foundation, the Task Exxon. Force began its work by organizing a number of focus groups. The AMS has now received a second grant of Council Congratulates Exxon $50,000 from the Exxon Education Foundation, as well as a grant of $165,000 from the National Science Foundation. Education Foundation For further information about the work of the Task Force, see “Building Excellence in Doctoral Mathematics De- At the Summer Mathfest in Burlington in August, the AMS partments”, Notices, November/December 1995, pages Council passed a resolution congratulating the Exxon Ed- 1170–1171. ucation Foundation on its fortieth anniversary. AMS Pres- ident Cathleen Morawetz presented the resolution during —Timothy Goggins, AMS Development Officer the awards banquet to Edward Ahnert, president of the Exxon Education Foundation, and to Robert Witte, senior program officer with Exxon. -
Leonard Eugene Dickson
LEONARD EUGENE DICKSON Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was a towering mathematician and inspiring teacher. He directed fifty-three doctoral dissertations, including those of 15 women. He wrote 270 papers and 18 books. His original research contributions made him a leading figure in the development of modern algebra, with significant contributions to number theory, finite linear groups, finite fields, and linear associative algebras. Under the direction of Eliakim Moore, Dickson received the first doctorate in mathematics awarded by the University of Chicago (1896). Moore considered Dickson to be the most thoroughly prepared mathematics student he ever had. Dickson spent some time in Europe studying with Sophus Lie in Leipzig and Camille Jordan in Paris. Upon his return to the United States, he taught at the University of California in Berkeley. He had an appointment with the University of Texas in 1899, but at the urging of Moore in 1900 he accepted a professorship at the University of Chicago, where he stayed until 1939. The most famous of Dickson’s books are Linear Groups with an Expansion of the Galois Field Theory (1901) and his three-volume History of the Theory of Numbers (1919 – 1923). The former was hailed as a milestone in the development of modern algebra. The latter runs to some 1600 pages, presenting in minute detail, a nearly complete guide to the subject from Pythagoras (or before 500 BCE) to about 1915. It is still considered the standard reference for everything that happened in number theory during that period. It remains the only source where one can find information as to who did what in various number theory topics. -
DICKSON's HISTORY OP the THEORY of NUMBERS. History Of
1919.] DICKSON ON THEOKY OF NUMBERS. 125 DICKSON'S HISTORY OP THE THEORY OF NUMBERS. History of the Theory of Numbers. By LEONARD EUGENE DICKSON. Volume I. Carnegie Institution of Washington, 1919. 486 pp. IN these days when "pure" science is looked upon with impatience, or at best with good-humored indulgence, the appearance of such a book as this will be greeted with joy by those of us who still believe in mathematics for mathematics' sake, as we do in art for art's sake or for music for the sake of music. For those who can see no use or importance in any studies in mathematics which do not smack of the machine shop or of the artillery field it may be worth while to glance through the index of authors in this volume and note the frequent appearance of names of men whose work in the "practical applications" of mathematics would almost qualify them for a place on the faculty of our most advanced educational institutions. One of the most assiduous students of the theory of numbers was Euler, whose work otherwise was of sufficient importance to attract the notice of a king of Prussia. The devotee of this peculiar branch of science can reassure himself that he is in very good company when he reads the list of authors cited in connection with the famous theorems of Fermât and Wilson in Chapter III; Cauchy, Cayley, d'Alembert, Dedekind, Euler, Gauss, Jacobi, Kron- ecker, Lagrange, Laplace, Legendre, Leibniz, Steiner, Syl vester, Von Staudt and a host of others, great and small, living and dead, to the number of over two hundred, who have found these absolutely "useless" theorems worthy of their most serious attention. -
An Outline of the History of Mathematics in the U
The Hilbert American Colony This file provides more biographical details on the Hilbert American colony than are given in the Transition_1900 section. Table 1 lists the 13 American students who earned doctorates under the direction of David Hilbert at Göttingen, in chronological order. Only one finished before the turn of the 20th century. One of the next two was Hilbert’s first female student. Rather than describing the lives and careers of the remaining eleven members of Hilbert’s American colony in chronological order, they appear according to their rankings published in the first three editions of American Men of Science (AMoS), which appeared in 1906, 1910 and 1921. In each of the three lists, scientists regarded as the most eminent in their field were asked to rank others in the field, with the top 1000 awarded stars beside their name. Eighty mathematicians were starred, and the numbers in the AMoS column in Table 1 designate the edition in which they were listed (if at all): 1 for 1906, 2 for 1910, and 3 for 1921. Name Year AMoS Legh Wilber Reid 1899 Anne Lucy Bosworth 1900 Edgar Jerome Townsend 1900 3 Earle Raymond Hedrick 1901 1, 2, 3 Charles Albert Noble 1901 Oliver Dimon Kellogg 1902 2, 3 Charles Max Mason 1903 2, 3 Wilhelmus David Westfall 1905 David Clinton Gillespie 1906 William DeWeese Cairns 1907 Arthur Robert Crathorne 1907 Charles Haseman 1907 Wallie Hurwitz 1910 3 Table 1: Hilbert’s American student Legh Wilber Reid (1867-1961) was the only American to complete requirements for a doctorate before 1900. -
M\" Obius Polynomials
M¨obiusPolynomials Will Murray∗ California State University, Long Beach Long Beach, CA 98040-1001 [email protected] December 11, 2013 What do bracelets, juggling patterns, and irreducible polynomials over finite fields have in common? We can count all of them using the same techniques. In this paper, we identify a set of polynomials that unify all the counting problems above. Despite their wide application, they have not received an explicit name in the literature. We will christen them M¨obiuspolynomials and explore some of their properties. We will highlight the role that M¨obiuspolynomials have played in the contexts above and then use them to extend a classic combinatorial proof from Fermat's little theorem to Euler's totient theorem. In the first section below, we will define our polynomials and derive some key facts about them. After a brief digression to enjoy the graphs of our polynomials in the complex plane, we will see that Mn(x) gives the number of aperiodic bracelets of length n that can be built using x possible types of gems. An immediate corollary will be that Mn(x) ≡ 0 (mod n) for all x 2 Z. In three subsequent sections, we will apply our polynomials to count juggling patterns, to count irreducible polynomials over finite fields, and to prove Euler's totient theorem. 1 Definition and properties To construct our polynomial, we first recall the M¨obius µ function defined on a e1 er positive integer n = p1 ··· pr , where the pi are distinct primes: arXiv:1312.3848v1 [math.CO] 13 Dec 2013 µ(1) := 1 ( (−1)r if all e = 1 e1 er i µ (n = p1 ··· pr ) := 0 if any ei > 1 ∗I thank Art Benjamin and Robert Mena for suggesting this investigation and many useful subsequent thoughts. -
Appendices Due to Concerns Over the Quality of the Data Collected
APPENDIX A WSU 2014-19 STRATEGIC PLAN Appendix A: WSU Strategic Plan 2014-15 Strategic Plan 2014-2019 President Elson S. Floyd, Ph.D. Strategic Plan 2014-2019 Introduction The 2014-19 strategic plan builds on the previous five-year plan, recognizing the core values and broad mission of Washington State University. Goals and strategies were developed to achieve significant progress toward WSU’s aspiration of becoming one of the nation’s leading land-grant universities, preeminent in research and discovery, teaching, and engagement. The plan emphasizes the institution’s unique role as an accessible, approachable research institution that provides opportunities to an especially broad array of students while serving Washington state’s broad portfolio of social and economic needs. While providing exceptional leadership in traditional land-grant disciplines, Washington State University adds value as an integrative partner for problem solving due to its innovative focus on applications and its breadth of program excellence. The plan explicitly recognizes the dramatic changes in public funding that have occurred over the duration of the previous strategic plan, along with the need for greater institutional nimbleness, openness, and entrepreneurial activity that diversifies the University’s funding portfolio. In addition, the plan reaffirms WSU’s land-grant mission by focusing greater attention system-wide on increasing access to educational opportunity, responding to the needs of Washington state through research, instruction, and outreach, and contributing to economic development and public policy. While the new plan retains the four key themes of the previous plan, its two central foci include offering a truly transformative educational experience to undergraduate and graduate students and accelerating the development of a preeminent research portfolio.