The 2005 Notices Index
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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. -
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 -
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. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
MSRI Celebrates Its Twentieth Birthday, Volume 50, Number 3
MSRI Celebrates Its Twentieth Birthday The past twenty years have seen a great prolifera- renewed support. Since then, the NSF has launched tion in mathematics institutes worldwide. An in- four more institutes: the Institute for Pure and spiration for many of them has been the Applied Mathematics at the University of California, Mathematical Sciences Research Institute (MSRI), Los Angeles; the AIM Research Conference Center founded in Berkeley, California, in 1982. An es- at the American Institute of Mathematics (AIM) in tablished center for mathematical activity that Palo Alto, California; the Mathematical Biosciences draws researchers from all over the world, MSRI has Institute at the Ohio State University; and the distinguished itself for its programs in both pure Statistical and Applied Mathematical Sciences and applied areas and for its wide range of outreach Institute, which is a partnership of Duke University, activities. MSRI’s success has allowed it to attract North Carolina State University, the University of many donations toward financing the construc- North Carolina at Chapel Hill, and the National tion of a new extension to its building. In October Institute of Statistical Sciences. 2002 MSRI celebrated its twentieth year with a Shiing-Shen Chern, Calvin C. Moore, and I. M. series of special events that exemplified what MSRI Singer, all on the mathematics faculty at the Uni- has become—a focal point for mathematical culture versity of California, Berkeley, initiated the original in all its forms, with the discovery and delight of proposal for MSRI; Chern served as the founding new mathematical knowledge the top priority. director, and Moore was the deputy director. -
Presidential Reflections: Interview with David Eisenbud
Presidential Reflections: Interview with David Eisenbud There is also the Committee on Committees, Every other year, when a new AMS president takes office, the which helps the president do this, because there Notices publishes interviews with the incoming and outgoing are something like 300 appointments a year that president. What follows is an edited version of an interview have to be made. My first act as president—really with David Eisenbud, whose two-year term as president ends as president-elect—was to gather together people on January 31, 2005. The interview was conducted in fall 2004 who I thought would be very well connected and by Notices senior writer and deputy editor Allyn Jackson. also who would reach into many different popu- Eisenbud is director of the Mathematical Sciences Research Institute (MSRI) in Berkeley and professor of mathematics at lations of mathematicians. One of my ambitions was the University of California, Berkeley. to provide a diverse new group of committee mem- An interview with AMS president-elect James Arthur will bers—young people and people from the minority appear in the March 2005 issue of the Notices. community. I also tried hard to make sure that women are well represented on committees and slates for elections. And I am proud of what we did Notices: The president of the AMS has two types in that respect. That’s actually the largest part of of duties. One type consists of the things that he or the president’s job, in terms of just sheer time and she has to do, by virtue of the office. -
Plenary Speakers
FoCM95 Park City: Plenary speakers: WEEK 1 Marie-Francoise Roy, Universite de Rennes Shmuel Winograd, IBM Dima Y. Grigoriev, Pennsylvania State University Richard S. Varga, Kent State University Steve Smale, University of California, Berkeley John CannyUniversity of California, Berkeley Felipe Cucker, Universitat Pampeu Fabra, Spain Victor Pan, Herbert H. Lehman College, CUNY Michael Shub, IBM Roger Brockett, Harvard University WEEK 2 Henryk Wozniakowski, University of Warsaw David Donoho, University of California, Berkeley and Columbia University Yosef Yomdin, Weizmann Institute of Science, Israel Margaret H. Wright, AT&T Bell Laboratories N. Karmarker, AT&T Bell Laboratories Manuel Blum, University of California, Berkeley Roger Temam, Indiana University Arkadi Nemirovski, Israel Institute of Technology Hubertus Th.Jongen, Reinisch-Westf Tech Hochschule James M. Renegar, Cornell University WEEK 3 Herb Keller, California Institute of Technology Gene H. Golub, Stanford University Alexandre J. Chorin, University of California, Berkeley T. Y. Li, Michigan State University James Yorke, University of Maryland Lenore Blum, MSRI Eugene L. Allgower, Colorado State University Arieh Iserles, University of Cambridge, UK James W. Demmel, University of California, Berkeley W. Dahmen, Reinisch-Westf Tech Hochschule WEEK 4 Ronald A. DeVore, University of South Carolina, Columbia Ulrich Kulisch, University of Karlsruhe Victor A. V. Vassiliev, Institute for System Studies, Moscow Jacques Louis Lions, College de France Henryk Wozniakowski, University of -
ON the ACTION of SO(3) on Sn
Pacific Journal of Mathematics ON THE ACTION OF SO.3/ ON Sn DEANE MONTGOMERY AND HANS SAMELSON Vol. 12, No. 2 February 1962 ON THE ACTION OF SO(3) ON Sn DEANE MONTGOMERY AND HANS SAMELSON 1. Introduction* This paper contains some facts about the possible actions of the rotation group SO(3) on the ^-sphere Sn. For some of the results the action is required to be differentiate, but for others this is not necessary. We recall for an action of a compact Lie group, that a principal isotropy group is an isotropy group of the lowest possible dimension and that among these it is one with the fewest possible components. An orbit with such an isotropy group is called a principal orbit. For a compact Lie group acting on a cohomology manifold over Z, principal orbits form an open connected everywhere dense set. In fact the comple- ment is a closed set of dimension at most n — 2 [1, Chapter IX]. Two of the results to be proved are the following, where B is the set of points on orbits of dimension less than the highest dimension of any orbit: If SO(3) acts differentiably on Sn with three-dimensional principal orbits and if dim B < n — 2 then the principal isotropy group is the identity; if SO(3) acts differentially on S7, then some orbit has dimension less than 3. Part of the motivation for our work was the attempt to discover whether the latter result is true for all n. If SO(3) does act differ- entiably on Sn with all orbits three-dimensional then, as far as rational coefficients are concerned, the sheaf generated by the orbits is constant and relative to these coefficients we obtain similar results to those for a fibering by S3 (see [1, 3]); hence n — 4/b — 1.