2018 Virginia Math Bulletin
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Of the American Mathematical Society August 2017 Volume 64, Number 7
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves -
M-Adic Perturbations in Noetherian Local Rings
Perturbations in Noetherian Local Rings m-adic Perturbations in Noetherian Local Rings Nick Cox-Steib [email protected] University of Missouri Abstract We develop new methods to study m-adic stability in an arbitrary Noetherian local ring. These tech- niques are used to prove results about the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine m-adic perturbations. 1 Introduction The topic of this paper involves comparing a fixed ideal, I =( f1,..., fc) = f , inside a Noetherian local ring, (R,m ), with ideals of the form ( f + ε , f + ε ,..., f + ε ) = f + ε , where ε ,...,ε are in R 1 1 2 2 c c 1 c some large power of the maximal ideal of R. When it is in the interest of clarity and space and the ideal I is understood, we often use the notation M M M M := = , and M := IM f M ( f +ε) f + ε M where M is an R-module. We will refer to the ideals f + ε , as mR-adic perturbations of I, and, more generally, we may also speak of the quotients M as perturbations of M. At the most basic level, we ( f +ε) want to understand the relationship between I and its perturbations. Recent interest in these problems stems from their relationship to the deformation of properties. This arXiv:2011.08044v2 [math.AC] 14 Dec 2020 relationship has been formalized in a recent paper by De Stefani and Smirnov with the concept of m-adic stability [SS20]. A property, P, of Noetherian local rings is m-adically stable if it is stable under sufficiently small perturbations of ideals generated by a regular element — that is, for every regular element f in a Noetherian local ring (R,m) such that R/( f ) has property P, there is an N ∈ N such that R/( f + ε) also has property P for all ε ∈ mN. -
Algebra & Number Theory Vol. 7 (2013)
Algebra & Number Theory Volume 7 2013 No. 3 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Edward Frenkel University of California, Berkeley, USA Michael Singer North Carolina State University, USA Andrew Granville Université de Montréal, Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Virginia, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University, -
Commutative Algebra Provides a Big Surprise for Craig Huneke's Birthday
CÓÑÑÙØaØiÚe aÐgebÖa ÔÖÓÚide× a big ×ÙÖÔÖi×e fÓÖ CÖaig ÀÙÒeke³× biÖØhdaÝ Irena Swanson Communicated by Tom Garrity A commutative algebra conference in July, 2016, on the occasion of Craig Huneke’s 65th birthday, included a major mathematical surprise. Craig Huneke has been at the forefront of research in commutative algebra, introducing and advancing several influential notions, such as d-sequences, licci ideals, symbolic powers, homological methods, computational methods, tight closure, uniform bounds, and prime characteristic methods. He has mentored 24 PhD students and many postdocs; he has co-organized conferences, such as the Kansas-Missouri-Nebraska KUMUNU commutative algebra conference; he has served on the Executive Committee of the AMS and on the Board of Trustees of the MSRI. Almost all the talks were directly related to Huneke’s work, and most speakers started their talks describing how Huneke affected their work as a mathematician and as a person; they talked about his mathematical productivity, extensive collaborations, productive and precious lunches with napkin notes, excellent and influential talks, his advising, mentoring, his friendly competitiveness, and so on. Claudia Polini addressed how commutative algebra in general is friendly to women (possibly due to Emmy Noether being one of us), and in particular how Huneke has been a tremendous role model as a teacher, collaborator, mentor, and organizer. His own family life, with wife Edith Clowes, Professor of Slavic Languages and Literatures, and their children Sam and Ned, has been a shining example for possibilities in home and professional life for women in academia. During the banquet, Hochster read his poem honoring Huneke. -
![Arxiv:1805.00492V2 [Math.AC] 12 Apr 2019 a Nt Rjciedimension](https://docslib.b-cdn.net/cover/5839/arxiv-1805-00492v2-math-ac-12-apr-2019-a-nt-rjciedimension-895839.webp)
Arxiv:1805.00492V2 [Math.AC] 12 Apr 2019 a Nt Rjciedimension
NON-COMMUTATIVE RESOLUTIONS OF TORIC VARIETIES ELEONORE FABER, GREG MULLER, AND KAREN E. SMITH Abstract. Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring EndR(A), where A is the (finite) direct sum of all (isomorphism classes of) conic R-modules, has finite global dimension equal to the dimension of R. This gives a precise version, and an elementary proof, of a theorem of Spenkoˇ and Van den Bergh implying that EndR(A) has finite global dimension. Furthermore, we show that EndR(A) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field k of prime charac- teristic, we show that the ring of differential operators Dk(R) has finite global dimension. 1. Introduction Consider a local or graded ring R which is commutative and Noetherian. A well-known theorem of Auslander-Buchsbaum and Serre states that R is regular if and only if R has finite global dimension—that is, if and only if every R-module has finite projective dimension. While the standard definition of regularity does not extend to non-commutative rings, the definition of global dimension does, so this suggests that finite global di- mension might play the role of regularity for non-commutative rings. This is an old idea going back at least to Dixmier [Dix63], though since then our understanding of connection between regularity and finite global dimension has been refined by the works of Auslander, Artin, Shelter, Van den Bergh and others. -
In the Supreme Court of Pennsylvania League Of
Received 2/15/2018 11:52:50 PM Supreme Court Middle District Filed 2/15/2018 11:52:00 PM Supreme Court Middle District 159 MM 2017 IN THE SUPREME COURT OF PENNSYLVANIA LEAGUE OF WOMEN VOTERS OF : PENNSYLVANIA, et al., : : Petitioners, : v. : Docket No. 159 MM 2017 : THE COMMONWEALTH OF : PENNSYLVANIA, et al., : : Respondents. : STATEMENT OF RESPONDENT THOMAS W. WOLF IN SUPPORT OF HIS PROPOSED REMEDIAL CONGRESSIONAL MAP PURSUANT TO COURT’S ORDERS OF JANUARY 22 AND JANUARY 26, 2018 TABLE OF CONTENTS I. The Governor’s Proposed Map Is Fair, Constitutional, and Respects the Criteria Set Forth by the Court. ............................................ 4 A. The Governor’s Map Respects Traditional Districting Principles ............................................................................................... 4 B. Statistical Analysis Underscores the Key Attributes of the Governor’s Map .................................................................................. 12 1. Splits of Political Subdivisions ................................................. 12 2. Compactness Scores .................................................................. 12 C. Mathematical Analysis Demonstrates That the Governor’s Map, in Contrast to Legislative Respondents’ Map, Gives Pennsylvania Voters a Fair and Unbiased Opportunity to Participate in Congressional Elections. ............................................... 13 II. This Court Has the Authority – and, Indeed, the Responsibility -- to Adopt a Remedial Map. ......................................................................... -
Lectures on Local Cohomology
Contemporary Mathematics Lectures on Local Cohomology Craig Huneke and Appendix 1 by Amelia Taylor Abstract. This article is based on five lectures the author gave during the summer school, In- teractions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Grothendieck concerning the connectedness of the spectrum of certain rings. Several applications are given, including a theorem of Fulton and Hansen concern- ing the connectedness of intersections of algebraic varieties. In an appendix written by Amelia Taylor, an another application is given to prove a theorem of Kalkbrenner and Sturmfels about the reduced initial ideals of prime ideals. Contents 1. Introduction 1 2. Local Cohomology 3 3. Injective Modules over Noetherian Rings and Matlis Duality 10 4. Cohen-Macaulay and Gorenstein rings 16 d 5. Vanishing Theorems and the Structure of Hm(R) 22 6. Vanishing Theorems II 26 7. Appendix 1: Using local cohomology to prove a result of Kalkbrenner and Sturmfels 32 8. Appendix 2: Bass numbers and Gorenstein Rings 37 References 41 1. Introduction Local cohomology was introduced by Grothendieck in the early 1960s, in part to answer a conjecture of Pierre Samuel about when certain types of commutative rings are unique factorization 2000 Mathematics Subject Classification. Primary 13C11, 13D45, 13H10. Key words and phrases. local cohomology, Gorenstein ring, initial ideal. The first author was supported in part by a grant from the National Science Foundation, DMS-0244405. -
The Koszul Homology of an Ideal
ADVANCES IN MATHEMATICS 56, 2955318 (1985) The Koszul Homology of an Ideal CRAIG HUNEKE* Junior Fellow, Michigan Society of’ Fellows. Uniuersity of Michigan, Ann Arbor, Michigan 48109 INTRODUCTION Recently many researchers have worked on problems connected with various graded algebras associated to an ideal I in a local ring R. Two algebras in particular have received the most attention: the associated graded algebra of Z, gr,( R) = R/Z@ Z/Z’@ ... , and the Rees algebra of Z, defined to be R[Zt]. Brodmann [2] and Goto and Shimoda [9] have studied the local cohomology of R[Zr] for certain primary ideals Z, Herzog has obtained new results in the case where Z is the maximal ideal, Eisenbud and Huneke [7] studied the CohenMacaulayness of these algebras, while the concepts of d-sequences [ 111 and Hodge algebras [S] have been used to understand these graded algebras. Recently Simis and Vasconcelos w, 211 related the Cohen Macaulayness, torsion-freeness, and normality of these algebras to the Koszul homology of I. If Z is an ideal, we let ZZ,(Z;R) denote the jth Koszul homology of the ideal Z with respect to somejixed system of generators for I. We denote the symmetric algebra of a module A4 by Sym(M), and denote thejth graded piece of this algebra by Sym,(M). In [20] and [21] Simis and Vasconcelos construct a complex A’(Z) (or simply A) with the following properties: (1) ,.K is graded, and if we let ,A?‘I”be the nth graded piece of A, then A” is the complex, +ff,(z; R)OSym,, ,(F/ZF)~H,~,(Z;R)OSY~,~~+,(F/ZF) -+ . -
Tight Closure and Elements of Small Order in Integral Extensions*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 71 (1991) 233-247 233 North-Holland Tight closure and elements of small order in integral extensions* Melvin Hochster Mathematics Department, University of Michigan, Awt Arbor, MI 48109-1003, USA Craig Huneke Mathematics Department, Purdue university, West Lafayette, IN 47907, USA Communicated by T. Hibi Received 7 September 1989 Dedicated to Professor Hideyuki Matsumura on his sixtieth birthday 1. Introduction Throughout this paper ‘ring’ means commutative ring with identity and modules are unital. Given rings are usually assumed to be Noetherian, but there are notable exceptions. The phrase ‘characteristic p’ always means ‘positive prime characteristic p’, and the letters q, q’, etc. denote pe, p”, etc., where e, e’ E N, the nonnegative integers. The authors have recently introduced the notion of tight closure for a sub- module N of a finitely generated module M over a Noetherian ring R of characteristic p and in certain equicharacteristic zero cases, including affine algebras over fields of characteristic 0. The theory started with the study of the notion of tight closure for an ideal I c R, i.e. with the case M = R, N = I,and this is still perhaps the most important case. The notion of tight closure has yielded new proofs, and, in many instances, unexpectedly strong improvements, of the local homological conjectures, of the existence of big Cohen-Macaulay modules, of the Cohen-Macaulay property for subrings which are direct summands of regular rings (where A is a direct summand of R means that A is a direct summand of R as an A-module) in the equicharacteristic case (in fact, tight closure techniques give the first proof of this fact in complete generality in the * Both authors were partially supported by grants from the National Science Foundation. -
A Computational Approach to Measuring Vote Elasticity and Competitiveness
A Computational Approach to Measuring Vote Elasticity and Competitiveness Daryl DeFord∗ CSAIL, Massachusetts Institute of Technology Moon Duchin Department of Mathematics, Tufts University Justin Solomon CSAIL, Massachusetts Institute of Technology May 27, 2020 Abstract The recent wave of attention to partisan gerrymandering has come with a push to refine or replace the laws that govern political redistricting around the country. A common element in several states' reform efforts has been the inclusion of com- petitiveness metrics, or scores that evaluate a districting plan based on the extent to which district-level outcomes are in play or are likely to be closely contested. In this paper, we examine several classes of competitiveness metrics motivated by recent reform proposals and then evaluate their potential outcomes across large ensembles of districting plans at the Congressional and state Senate levels. This is part of a growing literature using MCMC techniques from applied statistics to situate plans and criteria in the context of valid redistricting alternatives. Our empirical analysis focuses on five states|Utah, Georgia, Wisconsin, Virginia, and Massachusetts|chosen to represent a range of partisan attributes. We highlight arXiv:2005.12731v1 [cs.CY] 26 May 2020 situation-specific difficulties in creating good competitiveness metrics and show that optimizing competitiveness can produce unintended consequences on other partisan metrics. These results demonstrate the importance of (1) avoiding writing detailed metric constraints into long-lasting constitutional reform and (2) carrying out careful mathematical modeling on real geo-electoral data in each redistricting cycle. Keywords: Redistricting, Gerrymandering, Markov Chains, Competitiveness ∗The authors gratefully acknowledge the generous support of the Prof. -
CURRICULUM VITAE for IRENA SWANSON July
CURRICULUM VITAE FOR IRENA SWANSON July 2020 Irena Swanson Purdue University Office telephone: (765) 494-1909 150 N. University Street E-mail address: [email protected] West Lafayette, IN 47907 Web site: https://www.math.purdue.edu/~iswanso/ Education: Ph.D., Purdue University, 1992 B.A., Reed College, 1987 Career history: Professor and Department Head, Purdue University, July 2020- Professor, Reed College, 2005-June 2020, chair of the Mathematics Department, 2013/14, 2014/15. Fulbright-NAWI Graz Visiting Professor in the Natural Sciences, Graz, Austria, Fall 2018 Visiting Professor at University of Rome III, Italy, March 2010–May 2010 Visiting Professor at University of Ljubljana, Slovenia, Fall 2009 Professor, New Mexico State University, 2005–07 Mathematical Sciences Research Institute (MSRI), Berkeley, California, 2002–2003 Visiting Professor, University of Kansas, 2000–2001 Associate Professor, New Mexico State University, 2000–2005 Visiting Professor, University of L’Aquila, Italy, May-June 1999 Postdoctoral fellowship at MSRI, Berkeley, California, Fall 1998 Assistant Professor, New Mexico State University, 1995–2000 T. H. Hildebrandt Assistant Professor, University of Michigan, 1992–1995 Graduate teaching and research assistant, Purdue University, 1987–92 Honors: Fellow of the American Mathematical Society, Class of 2019. Fulbright Fellowship, NAWI Graz, Austria, Fall 2018. Purdue University Mathematics Department Outstanding Alumna for 2007–08. Member of: American Mathematical Society Association for Women in Mathematics Mathematical Association of America Druˇstvo matematikov, fizikov in astronomov Slovenije Phi Beta Kappa Ph.D. students: Ibrahim Al-Ayyoub, 2004, New Mexico State University Rebecca Pablo Garc´ıa, 2004, New Mexico State University Mark Rhodes, 2001, New Mexico State University Co-advised Master’s and Ph.D. -
Algebra & Number Theory
Algebra & Number Theory Volume 4 2010 No. 2 mathematical sciences publishers Algebra & Number Theory www.jant.org EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Andrei Okounkov Princeton University, USA John H. Coates University of Cambridge, UK Raman Parimala Emory University, USA J-L. Colliot-Thel´ ene` CNRS, Universite´ Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hel´ ene` Esnault Universitat¨ Duisburg-Essen, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universitat,¨ Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Ronald Solomon Ohio State University, USA Andrew Granville Universite´ de Montreal,´ Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Kansas, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale