Biennial Overview of AMS Honors
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Questions About Boij-S\" Oderberg Theory
QUESTIONS ABOUT BOIJ–SODERBERG¨ THEORY DANIEL ERMAN AND STEVEN V SAM 1. Background on Boij–Soderberg¨ Theory Boij–S¨oderberg theory focuses on the properties and duality relationship between two types of numerical invariants. One side involves the Betti table of a graded free resolution over the polynomial ring. The other side involves the cohomology table of a coherent sheaf on projective space. The theory began with a conjectural description of the cone of Betti tables of finite length modules, given in [10]. Those conjectures were proven in [25], which also described the cone of cohomology tables of vector bundles and illustrated a sort of duality between Betti tables and cohomology tables. The theory itself has since expanded in many directions: allowing modules whose support has higher dimension, replacing vector bundles by coherent sheaves, working over rings other than the polynomial ring, and so on. But at its core, Boij–S¨oderberg theory involves: (1) A classification, up to scalar multiple, of the possible Betti tables of some class of objects (for example, free resolutions of finitely generated modules of dimension ≤ c). (2) A classification, up to scalar multiple, of the cohomology tables of some class of objects (for examples, coherent sheaves of dimension ≤ n − c). (3) Intersection theory-style duality results between Betti tables and cohomology tables. One motivation behind Boij and S¨oderberg’s conjectures was the observation that it would yield an immediate proof of the Cohen–Macaulay version of the Multiplicity Conjectures of Herzog–Huneke–Srinivasan [44]. Eisenbud and Schreyer’s [25] thus yielded an immediate proof of that conjecture, and the subsequent papers [11, 26] provided a proof of the Mul- tiplicity Conjecture for non-Cohen–Macaulay modules. -
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. -
The Geometry of Syzygies
The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . -
Locally Compact Groups: Traditions and Trends Karl Heinrich Hofmann Technische Universitat Darmstadt, [email protected]
University of Dayton eCommons Summer Conference on Topology and Its Department of Mathematics Applications 6-2017 Locally Compact Groups: Traditions and Trends Karl Heinrich Hofmann Technische Universitat Darmstadt, [email protected] Wolfgang Herfort Francesco G. Russo Follow this and additional works at: http://ecommons.udayton.edu/topology_conf Part of the Geometry and Topology Commons, and the Special Functions Commons eCommons Citation Hofmann, Karl Heinrich; Herfort, Wolfgang; and Russo, Francesco G., "Locally Compact Groups: Traditions and Trends" (2017). Summer Conference on Topology and Its Applications. 47. http://ecommons.udayton.edu/topology_conf/47 This Plenary Lecture is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted for inclusion in Summer Conference on Topology and Its Applications by an authorized administrator of eCommons. For more information, please contact [email protected], [email protected]. Some Background Notes Some \new" tools Near abelian groups Applications Alexander Doniphan Wallace (1905{1985) Gordon Thomas Whyburn Robert Lee Moore Some Background Notes Some \new" tools Near abelian groups Applications \The best mathematics is the most mixed-up mathematics, those disciplines in which analysis, algebra and topology all play a vital role." Gordon Thomas Whyburn Robert Lee Moore Some Background Notes Some \new" tools Near abelian groups Applications \The best mathematics is the most mixed-up mathematics, those disciplines in which -
S0002-9904-1917-02959-X.Pdf
1917.] EMORY McCLINTOCK. 353 £ = *(*), * = *«). It is clear that corresponding to any point w in the vicinity of f(zo) the function z = ^(£) furnishes n values of z. Also the form of ^(£) would depend on the particular circle chosen, but one form may be transformed into any other by replacing £ by the product of £ and the appropriate nth root of unity. UNIVERSITY OF TORONTO. EMORY McCLINTOCK. BUT few members of the American Mathematical Society at the present time appreciate the magnitude of the services rendered by its former president, Emory McClintock, who died July 10, 1916. He was born September 19, 1840, at Carlisle, Pa. His father was the Rev. John McClintock, a learned Methodist Episcopal clergyman, for a time professor of mathematics, Latin, and Greek in Dickinson College, and during the Civil War chaplain of the American Chapel in Paris. He is perhaps best known as the author, with another, of a " Cyclopaedia of Biblical, Theological, and Ecclesiastical Literature." McClintock went to school for the first time at the age of thirteen, and a year later entered the freshman year of Dickin son College. In 1856, when his father left Dickinson College for New York, he entered Yale, and in 1857 he entered Colum bia as a member of the class of 1859. His remarkable ability excited the admiration of his teachers, Professors Charles Davies and William Guy Peck. In April, 1859 in order to meet an emergency caused by the illness of a member of the teaching staff, he was graduated and appointed tutor in mathe matics. Soon afterwards his father took charge of the Ameri can Chapel in Paris, and in 1860 McClintock resigned his position at Columbia to go abroad. -
Downloaded from Brill.Com09/24/2021 10:06:53AM Via Free Access 268 Revue De Synthèse : TOME 139 7E SÉRIE N° 3-4 (2018) Chercheur Pour IBM
REVUE DE SYNTHÈSE : TOME 139 7e SÉRIE N° 3-4 (2018) 267-288 brill.com/rds A Task that Exceeded the Technology: Early Applications of the Computer to the Lunar Three-body Problem Allan Olley* Abstract: The lunar Three-Body problem is a famously intractable problem of Newtonian mechanics. The demand for accurate predictions of lunar motion led to practical approximate solutions of great complexity, constituted by trigonometric series with hundreds of terms. Such considerations meant there was demand for high speed machine computation from astronomers during the earliest stages of computer development. One early innovator in this regard was Wallace J. Eckert, a Columbia University professor of astronomer and IBM researcher. His work illustrates some interesting features of the interaction between computers and astronomy. Keywords: history of astronomy – three body problem – history of computers – Wallace J. Eckert Une tâche excédant la technologie : l’utilisation de l’ordinateur dans le problème lunaire des trois corps Résumé : Le problème des trois corps appliqué à la lune est un problème classique de la mécanique newtonienne, connu pour être insoluble avec des méthodes exactes. La demande pour des prévisions précises du mouvement lunaire menait à des solutions d’approximation pratiques qui étaient d’une complexité considérable, avec des séries tri- gonométriques contenant des centaines de termes. Cela a très tôt poussé les astronomes à chercher des outils de calcul et ils ont été parmi les premiers à utiliser des calculatrices rapides, dès les débuts du développement des ordinateurs modernes. Un innovateur des ces années-là est Wallace J. Eckert, professeur d’astronomie à Columbia University et * Allan Olley, born in 1979, he obtained his PhD-degree from the Institute for the History and Philosophy of Science Technology (IHPST), University of Toronto in 2011. -
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. -
List of Members
LIST OF MEMBERS, ALFRED BAKER, M.A., Professor of Mathematics, University of Toronto, Toronto, Canada. ARTHUR LATHAM BAKER, C.E., Ph.D., Professor of Mathe matics, Stevens School, Hpboken., N. J. MARCUS BAKER, U. S. Geological Survey, Washington, D.C. JAMES MARCUS BANDY, B.A., M.A., Professor of Mathe matics and Engineering, Trinit)^ College, N. C. EDGAR WALES BASS, Professor of Mathematics, U. S. Mili tary Academy, West Point, N. Y. WOOSTER WOODRUFF BEMAN, B.A., M.A., Member of the London Mathematical Society, Professor of Mathe matics, University of Michigan, Ann Arbor, Mich. R. DANIEL BOHANNAN, B.Sc, CE., E.M., Professor of Mathematics and Astronomy, Ohio State University, Columbus, Ohio. CHARLES AUGUSTUS BORST, M.A., Assistant in Astronomy, Johns Hopkins University, Baltimore, Md. EDWARD ALBERT BOWSER, CE., LL.D., Professor of Mathe matics, Rutgers College, New Brunswick, N. J. JOHN MILTON BROOKS, B.A., Instructor in Mathematics, College of New Jersey, Princeton, N. J. ABRAM ROGERS BULLIS, B.SC, B.C.E., Macedon, Wayne Co., N. Y. WILLIAM ELWOOD BYERLY, Ph.D., Professor of Mathematics, Harvard University, Cambridge*, Mass. WILLIAM CAIN, C.E., Professor of Mathematics and Eng ineering, University of North Carolina, Chapel Hill, N. C. CHARLES HENRY CHANDLER, M.A., Professor of Mathe matics, Ripon College, Ripon, Wis. ALEXANDER SMYTH CHRISTIE, LL.M., Chief of Tidal Division, U. S. Coast and Geodetic Survey, Washington, D. C. JOHN EMORY CLARK, M.A., Professor of Mathematics, Yale University, New Haven, Conn. FRANK NELSON COLE, Ph.D., Assistant Professor of Mathe matics, University of Michigan, Ann Arbor, Mich. -
Wallace Eckert
Wallace Eckert Nakumbuka Dk Eckert aliniambia, "Siku moja, kila mtu atakuwa na kompyuta kwenye dawati lao." Macho yangu yalifunguka. Hiyo lazima iwe katika miaka mapema ya 1950’s. Aliona mapema. -Eleanor Krawitz Kolchin, mahojiano ya Huffington Post, Februari 2013. Picha: Karibu 1930, Jalada la Columbiana. Wallace John Eckert, 1902-1971. Pamoja na masomo ya kuhitimu huko Columbia, Chuo Kikuu cha Chicago, na Yale, alipokea Ph.D. kutoka Yale mnamo 1931 chini ya Profesa Ernest William Brown (1866-1938), ambaye alitumia kazi yake katika kuendeleza nadharia ya mwongozo wa mwezi. Maarufu zaidi kwa mahesabu ya mzunguko wa mwezi ambayo yaliongoza misheni ya Apollo kwenda kwa mwezi, Eckert alikuwa Profesa wa Sayansi ya Chuo Kikuu cha Columbia kutoka 1926 hadi 1970, mwanzilishi na Mkurugenzi wa Ofisi ya Taasisi ya Taaluma ya Thomas J. Watson katika Chuo Kikuu cha Columbia (1937-40), Mkurugenzi wa Ofisi ya Amerika ya US Naval Observatory Nautical Almanac (1940-45), na mwanzilishi na Mkurugenzi wa Maabara ya Sayansi ya Watson ya Sayansi katika Chuo Kikuu cha Columbia (1945-1966). Kwanza kabisa, na daima ni mtaalam wa nyota, Eckert aliendesha na mara nyingi alisimamia ujenzi wa mashine za kompyuta zenye nguvu kusuluhisha shida katika mechanics ya mbinguni, haswa ili kuhakikisha, kupanua, na kuboresha nadharia ya Brown. Alikuwa mmoja wa kwanza kutumia mashine za kadi za kuchomwa kwa suluhisho la shida tata za kisayansi. Labda kwa maana zaidi, alikuwa wa kwanza kusasisha mchakato wakati, mnamo 1933-34, aliunganisha mahesabu na kompyuta za IBM kadhaa na mzunguko wa vifaa na vifaa vya muundo wake ili kusuluhisha usawa wa aina, njia ambazo baadaye zilibadilishwa na kupanuliwa kwa IBM ya "Aberdeen "Calculator inayoweza kupatikana ya Udhibiti wa Mpangilio, Punch Kuhesabu elektroniki, Calculator ya Kadi iliyopangwa, na SSEC. -
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. -
President's Report
AWM ASSOCIATION FOR WOMEN IN MATHE MATICS Volume 36, Number l NEWSLETTER March-April 2006 President's Report Hidden Help TheAWM election results are in, and it is a pleasure to welcome Cathy Kessel, who became President-Elect on February 1, and Dawn Lott, Alice Silverberg, Abigail Thompson, and Betsy Yanik, the new Members-at-Large of the Executive Committee. Also elected for a second term as Clerk is Maura Mast.AWM is also pleased to announce that appointed members BettyeAnne Case (Meetings Coordi nator), Holly Gaff (Web Editor) andAnne Leggett (Newsletter Editor) have agreed to be re-appointed, while Fern Hunt and Helen Moore have accepted an extension of their terms as Member-at-Large, to join continuing members Krystyna Kuperberg andAnn Tr enk in completing the enlarged Executive Committee. I look IN THIS ISSUE forward to working with this wonderful group of people during the coming year. 5 AWM ar the San Antonio In SanAntonio in January 2006, theAssociation for Women in Mathematics Joint Mathematics Meetings was, as usual, very much in evidence at the Joint Mathematics Meetings: from 22 Girls Just Want to Have Sums the outstanding mathematical presentations by women senior and junior, in the Noerher Lecture and the Workshop; through the Special Session on Learning Theory 24 Education Column thatAWM co-sponsored withAMS and MAA in conjunction with the Noether Lecture; to the two panel discussions thatAWM sponsored/co-sponsored.AWM 26 Book Review also ran two social events that were open to the whole community: a reception following the Gibbs lecture, with refreshments and music that was just right for 28 In Memoriam a networking event, and a lunch for Noether lecturer Ingrid Daubechies.