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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. -
A Brief History of Edge-Colorings — with Personal Reminiscences
Discrete Mathematics Letters Discrete Math. Lett. 6 (2021) 38–46 www.dmlett.com DOI: 10.47443/dml.2021.s105 Review Article A brief history of edge-colorings – with personal reminiscences∗ Bjarne Toft1;y, Robin Wilson2;3 1Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark 2Department of Mathematics and Statistics, Open University, Walton Hall, Milton Keynes, UK 3Department of Mathematics, London School of Economics and Political Science, London, UK (Received: 9 June 2020. Accepted: 27 June 2020. Published online: 11 March 2021.) c 2021 the authors. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract In this article we survey some important milestones in the history of edge-colorings of graphs, from the earliest contributions of Peter Guthrie Tait and Denes´ Konig¨ to very recent work. Keywords: edge-coloring; graph theory history; Frank Harary. 2020 Mathematics Subject Classification: 01A60, 05-03, 05C15. 1. Introduction We begin with some basic remarks. If G is a graph, then its chromatic index or edge-chromatic number χ0(G) is the smallest number of colors needed to color its edges so that adjacent edges (those with a vertex in common) are colored differently; for 0 0 0 example, if G is an even cycle then χ (G) = 2, and if G is an odd cycle then χ (G) = 3. For complete graphs, χ (Kn) = n−1 if 0 0 n is even and χ (Kn) = n if n is odd, and for complete bipartite graphs, χ (Kr;s) = max(r; s). -
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. -
R Mathematics Esearch Eports
Mathematics r research reports M r Boris Hasselblatt, Svetlana Katok, Michele Benzi, Dmitry Burago, Alessandra Celletti, Tobias Holck Colding, Brian Conrey, Josselin Garnier, Timothy Gowers, Robert Griess, Linus Kramer, Barry Mazur, Walter Neumann, Alexander Olshanskii, Christopher Sogge, Benjamin Sudakov, Hugh Woodin, Yuri Zarhin, Tamar Ziegler Editorial Volume 1 (2020), p. 1-3. <http://mrr.centre-mersenne.org/item/MRR_2020__1__1_0> © The journal and the authors, 2020. Some rights reserved. This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Mathematics Research Reports is member of the Centre Mersenne for Open Scientific Publishing www.centre-mersenne.org Mathema tics research reports Volume 1 (2020), 1–3 Editorial This is the inaugural volume of Mathematics Research Reports, a journal owned by mathematicians, and dedicated to the principles of fair open access and academic self- determination. Articles in Mathematics Research Reports are freely available for a world-wide audi- ence, with no author publication charges (diamond open access) but high production value, thanks to financial support from the Anatole Katok Center for Dynamical Sys- tems and Geometry at the Pennsylvania State University and to the infrastructure of the Centre Mersenne. The articles in MRR are research announcements of significant ad- vances in all branches of mathematics, short complete papers of original research (up to about 15 journal pages), and review articles (up to about 30 journal pages). They communicate their contents to a broad mathematical audience and should meet high standards for mathematical content and clarity. The entire Editorial Board approves the acceptance of any paper for publication, and appointments to the board are made by the board itself. -
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. -
Journal Abbreviations
Abbreviations of Names of Serials This list gives the form of references used in Mathematical Reviews (MR). not previously listed ⇤ The abbreviation is followed by the complete title, the place of publication journal indexed cover-to-cover § and other pertinent information. † monographic series Update date: July 1, 2016 4OR 4OR. A Quarterly Journal of Operations Research. Springer, Berlin. ISSN Acta Math. Hungar. Acta Mathematica Hungarica. Akad. Kiad´o,Budapest. § 1619-4500. ISSN 0236-5294. 29o Col´oq. Bras. Mat. 29o Col´oquio Brasileiro de Matem´atica. [29th Brazilian Acta Math. Sci. Ser. A Chin. Ed. Acta Mathematica Scientia. Series A. Shuxue † § Mathematics Colloquium] Inst. Nac. Mat. Pura Apl. (IMPA), Rio de Janeiro. Wuli Xuebao. Chinese Edition. Kexue Chubanshe (Science Press), Beijing. ISSN o o † 30 Col´oq. Bras. Mat. 30 Col´oquio Brasileiro de Matem´atica. [30th Brazilian 1003-3998. ⇤ Mathematics Colloquium] Inst. Nac. Mat. Pura Apl. (IMPA), Rio de Janeiro. Acta Math. Sci. Ser. B Engl. Ed. Acta Mathematica Scientia. Series B. English § Edition. Sci. Press Beijing, Beijing. ISSN 0252-9602. † Aastaraam. Eesti Mat. Selts Aastaraamat. Eesti Matemaatika Selts. [Annual. Estonian Mathematical Society] Eesti Mat. Selts, Tartu. ISSN 1406-4316. Acta Math. Sin. (Engl. Ser.) Acta Mathematica Sinica (English Series). § Springer, Berlin. ISSN 1439-8516. † Abel Symp. Abel Symposia. Springer, Heidelberg. ISSN 2193-2808. Abh. Akad. Wiss. G¨ottingen Neue Folge Abhandlungen der Akademie der Acta Math. Sinica (Chin. Ser.) Acta Mathematica Sinica. Chinese Series. † § Wissenschaften zu G¨ottingen. Neue Folge. [Papers of the Academy of Sciences Chinese Math. Soc., Acta Math. Sinica Ed. Comm., Beijing. ISSN 0583-1431. -
German Jews in the United States: a Guide to Archival Collections
GERMAN HISTORICAL INSTITUTE,WASHINGTON,DC REFERENCE GUIDE 24 GERMAN JEWS IN THE UNITED STATES: AGUIDE TO ARCHIVAL COLLECTIONS Contents INTRODUCTION &ACKNOWLEDGMENTS 1 ABOUT THE EDITOR 6 ARCHIVAL COLLECTIONS (arranged alphabetically by state and then city) ALABAMA Montgomery 1. Alabama Department of Archives and History ................................ 7 ARIZONA Phoenix 2. Arizona Jewish Historical Society ........................................................ 8 ARKANSAS Little Rock 3. Arkansas History Commission and State Archives .......................... 9 CALIFORNIA Berkeley 4. University of California, Berkeley: Bancroft Library, Archives .................................................................................................. 10 5. Judah L. Mages Museum: Western Jewish History Center ........... 14 Beverly Hills 6. Acad. of Motion Picture Arts and Sciences: Margaret Herrick Library, Special Coll. ............................................................................ 16 Davis 7. University of California at Davis: Shields Library, Special Collections and Archives ..................................................................... 16 Long Beach 8. California State Library, Long Beach: Special Collections ............. 17 Los Angeles 9. John F. Kennedy Memorial Library: Special Collections ...............18 10. UCLA Film and Television Archive .................................................. 18 11. USC: Doheny Memorial Library, Lion Feuchtwanger Archive ................................................................................................... -
Contemporary Mathematics 78
CONTEMPORARY MATHEMATICS 78 Braids Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Artin's Braid Group held July 13-26. 1986 at the University of California, Santa Cruz, California Joan S. Birman Anatoly Libgober Editors http://dx.doi.org/10.1090/conm/078 Recent Titles in This Series 120 Robert S. Doran, Editor, Selfadjoint and nonselfadjoint operator algebras and operator theory, 1991 119 Robert A. Melter, Azriel Rosenfeld, and Prabir Bhattacharya, Editors, Vision geometry, 1991 118 Yan Shi-Jian, Wang Jiagang, and Yang Chung-chun, Editors, Probability theory and its applications in China, 1991 117 Morton Brown, Editor, Continuum theory and dynamical systems, 1991 116 Brian Harboume and Robert Speiser, Editors, Algebraic geometry: Sundance 1988, 1991 115 Nancy Flournoy an'il Robert K. Tsutakawa, Editors, Statistical multiple integration, 1991 114 Jeffrey C. Lagarias and Michael J. Todd, Editors, Mathematical developments arising from linear programming, 1990 113 Eric Grinberg and Eric Todd Quinto, Editors, Integral geometry and tomography, 1990 112 Philip J. Brown and Wayne A. Fuller, Editors, Statistical analysis of measurement error models and applications, 1990 Ill Earl S. Kramer and Spyros S. Magliveras, Editors, Finite geometries and combinatorial designs, I 990 II 0 Georgia Benkart and J. Marshall Osborn, Editors, Lie algebras and related topics, 1990 109 Benjamin Fine, Anthony Gaglione, and Francis C. Y. Tang, Editors, Combinatorial group theory, 1990 108 Melvyn S. Berger, Editor, Mathematics of nonlinear science, 1990 107 Mario Milman and Tomas Schonbek, Editors, Harmonic analysis and partial differential equations, 1990 I 06 Wilfried Sieg, Editor, Logic and computation, 1990 I 05 Jerome Kaminker, Editor, Geometric and topological invariants of elliptic operators, 1990 I 04 Michael Makkai and Robert Pare, Accessible categories: The foundations of categorical model theory, 1989 I 03 Steve Fisk, Coloring theories, 1989 I 02 Stephen McAdam, Primes associated to an ideal, 1989 101 S.-Y. -
Hyberbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves CBMS-NSF REGIONAL CONFERENCE SERIES in APPLIED MATHEMATICS
Hyberbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and published by SIAM. GARRETT BIRKHOFF, The Numerical Solution of Elliptic Equations D. V. LINDLEY, Bayesian Statistics, A Review R. S. VARGA, Functional Analysis and Approximation Theory in Numerical Analysis R. R. BAHADUR, Some Limit Theorems in Statistics PATRICK BILLINGSLEY, Weak Convergence of Measures: Applications in Probability J. L. LIONS, Some Aspects of the Optimal Control of Distributed Parameter Systems ROGER PENROSE, Techniques of Differential Topology in Relativity HERMAN CHERNOFF, Sequential Analysis and Optimal Design J. DURBIN, Distribution Theory for Tests Based on the Sample Distribution Function SOL I. RUBINOW, Mathematical Problems in the Biological Sciences P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves I. J. SCHOENBERG, Cardinal Spline Interpolation IVAN SINGER, The Theory of Best Approximation and Functional Analysis WERNER C. RHEINBOLDT, Methods of Solving Systems of Nonlinear Equations HANS F. WEINBERGER, Variational Methods for Eigenvalue Approximation R. TYRRELL ROCKAFELLAR, Conjugate Duality and Optimization SIR JAMES LIGHTHILL, Mathematical Biofluiddynamics GERARD SALTON, Theory of Indexing CATHLEEN S. MORAWETZ, Notes on Time Decay and Scattering for Some Hyperbolic Problems F. HOPPENSTEADT, Mathematical Theories of Populations: Demographics, Genetics and Epidemics RICHARD ASKEY, Orthogonal Polynomials and Special Functions L. E. PAYNE, Improperly Posed Problems in Partial Differential Equations S. ROSEN, Lectures on the Measurement and Evaluation of the Performance of Computing Systems HERBERT B. -
Notices of the American Mathematical
OF THE AMERICAN MATHEMATICAL SOCIETY VOLUMf 12, NUMBER 2 ISSUE NO. 80 FEBRUARY 1965 cNotiaiJ OF THE AMERICAN MATHEMATICAL SOCIETY Edited by John W. Green and Gordon L. \Yalker CONTENTS MEETINGS Calendar of Meetings o o o o o o o o o • o o • o o o o o o o • o 0 o 0 o 0 • 0 • 0 0 • 0 0 184 Program of the Meeting in New York. o 0 o 0 o 0 0 • 0 o o o 0 0 0 0 • 0 0 0 0 0 • 185 Abstracts for the Meeting- Pages 209-216 PRELIMINARY ANNOUNCEMENTS OF MEETINGS • o o o o o o o o • o 0 • 0 • 0 • • • 188 ACTIVITIES OF OTHER ASSOCIATIONS •••• o o. o o o o • o o o •• o 0 0 0. o 0 0 0 0 191 LETTERS TO THE EDITOR • o ••••• o o • o o o o o • o ••• o •••• o o o •• o o • o o 192 MEMORANDA TO MEMBERS List of Retired Mathematicians 190 Increase in Page Charges o o ••• o •• o • o o o •••••• 0 ••• 0 0 0 •• 0 0 0 0 194 Journals in Microform • o •••• o 0 ••• o • o ••• 0 •• o ••••• 0 •• 0 • • • • 194 Backlog of Mathematics Research Journals 0 0 0. 0. 0 ••• 0 •• 0 •• 0.. 195 Corporate Members ••••••••••••• 0 • 0 0 0 0 •• 0 0 • 0 •• 0 0 •• 0 ••• 0 201 ASSISTANTSHIPS AND FELLOWSHIPS IN MATHEMATICS 1965-1966 0 ••• 0 • 0 196 NEW AMS PUBLICATIONS • o •• o •• o •••• o • o o ••••• o o o •• o o •• o • o • o • 197 PERSONAL ITEMS .••••• o •• o. -
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. -
Morita Equivalence and Continuous-Trace C*-Algebras, 1998 59 Paul Howard and Jean E
http://dx.doi.org/10.1090/surv/060 Selected Titles in This Series 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D.