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Opening Ceremony
Opening ceremony Sir John Ball, President of the International Mathematical Union Your Majesty, Señor Ruiz Gallardón, Señora Cabrera, Señora Aguirre, Professor Manuel de León, Distinguished guests, Ladies and gentlemen, ¡Bienvenidos al ICM dos mil seis! Welcome to ICM 2006, the 25th International Congress of Mathematicians, and the first ICM to be held in Spain. We offer our heartfelt thanks to the Spanish nation, so rich in history and culture, for its invitation to Madrid. We greatly appreciate that His Majesty King Juan Carlos is honouring mathematics by His presence here today. While celebrating this feast of mathematics, with the many talking-points that it will provide, it is worth reflecting on the ways in which our community functions. Mathematics is a profession of high standards and integrity. We freely discuss our work with others without fear of it being stolen, and research is communicated openly prior to formal publication. Editorial procedures are fair and proper, and work gains its reputation through merit and not by how it is promoted. These are the norms operated by the vast majority of mathematicians. The exceptions are rare, and they are noticed. Mathematics has a strong record of service, freely given. We see this in the time and care spent in the refereeing of papers and other forms of peer review. We see it in the running of mathematical societies and journals, in the provision of free mathematical software and teaching resources, and in the various projects world-wide to improve electronic access to the mathematical literature, old and new. We see it in the nurturing of students beyond the call of duty. -
Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings
City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2005 Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings Mahdi Majidi-Zolbanin Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/1765 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings by Mahdi Majidi-Zolbanin A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of NewYork. 2005 UMI Number: 3187456 Copyright 2005 by Majidi-Zolbanin, Mahdi All rights reserved. UMI Microform 3187456 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 ii c 2005 Mahdi Majidi-Zolbanin All Rights Reserved iii This manuscript has been read and accepted for the Graduate Faculty in Mathematics in satisfaction of the dissertation requirements for the degree of Doctor of Philosophy. Lucien Szpiro Date Chair of Examining Committee Jozek Dodziuk Date Executive Officer Lucien Szpiro Raymond Hoobler Alphonse Vasquez Ian Morrison Supervisory Committee THE CITY UNIVERSITY OF NEW YORK iv Abstract Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings by Mahdi Majidi-Zolbanin Advisor: Professor Lucien Szpiro In this dissertation we study splitting of vector bundles of small rank on punctured spectrum of regular local rings. -
Depth, Dimension and Resolutions in Commutative Algebra
Depth, Dimension and Resolutions in Commutative Algebra Claire Tête PhD student in Poitiers MAP, May 2014 Claire Tête Commutative Algebra This morning: the Koszul complex, regular sequence, depth Tomorrow: the Buchsbaum & Eisenbud criterion and the equality of Aulsander & Buchsbaum through examples. Wednesday: some elementary results about the homology of a bicomplex Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) We would like to find something like this dm dm−1 d1 · · · Fm Fm−1 · · · F1 F0 A/a with A-modules Fi as simple as possible and s.t. Im di = Ker di−1. Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) We would like to find something like this dm dm−1 d1 · · · Fm Fm−1 · · · F1 F0 A/a with A-modules Fi as simple as possible and s.t. Im di = Ker di−1. We say that F· is a resolution of the A-module A/a Claire Tête Commutative Algebra I will begin with a little example. -
William M. Goldman June 24, 2021 CURRICULUM VITÆ
William M. Goldman June 24, 2021 CURRICULUM VITÆ Professional Preparation: Princeton Univ. A. B. 1977 Univ. Cal. Berkeley Ph.D. 1980 Univ. Colorado NSF Postdoc. 1980{1981 M.I.T. C.L.E. Moore Inst. 1981{1983 Appointments: I.C.E.R.M. Member Sep. 2019 M.S.R.I. Member Oct.{Dec. 2019 Brown Univ. Distinguished Visiting Prof. Sep.{Dec. 2017 M.S.R.I. Member Jan.{May 2015 Institute for Advanced Study Member Spring 2008 Princeton University Visitor Spring 2008 M.S.R.I. Member Nov.{Dec. 2007 Univ. Maryland Assoc. Chair for Grad. Studies 1995{1998 Univ. Maryland Professor 1990{present Oxford Univ. Visiting Professor Spring 1989 Univ. Maryland Assoc. Professor 1986{1990 M.I.T. Assoc. Professor 1986 M.S.R.I. Member 1983{1984 Univ. Maryland Visiting Asst. Professor Fall 1983 M.I.T. Asst. Professor 1983 { 1986 1 2 W. GOLDMAN Publications (1) (with D. Fried and M. Hirsch) Affine manifolds and solvable groups, Bull. Amer. Math. Soc. 3 (1980), 1045{1047. (2) (with M. Hirsch) Flat bundles with solvable holonomy, Proc. Amer. Math. Soc. 82 (1981), 491{494. (3) (with M. Hirsch) Flat bundles with solvable holonomy II: Ob- struction theory, Proc. Amer. Math. Soc. 83 (1981), 175{178. (4) Two examples of affine manifolds, Pac. J. Math.94 (1981), 327{ 330. (5) (with M. Hirsch) A generalization of Bieberbach's theorem, Inv. Math. , 65 (1981), 1{11. (6) (with D. Fried and M. Hirsch) Affine manifolds with nilpotent holonomy, Comm. Math. Helv. 56 (1981), 487{523. (7) Characteristic classes and representations of discrete subgroups of Lie groups, Bull. -
Reflexivity Revisited
REFLEXIVITY REVISITED MOHSEN ASGHARZADEH ABSTRACT. We study some aspects of reflexive modules. For example, we search conditions for which reflexive modules are free or close to free modules. 1. INTRODUCTION In this note (R, m, k) is a commutative noetherian local ring and M is a finitely generated R- module, otherwise specializes. The notation stands for a general module. For simplicity, M the notation ∗ stands for HomR( , R). Then is called reflexive if the natural map ϕ : M M M M is bijection. Finitely generated projective modules are reflexive. In his seminal paper M→M∗∗ Kaplansky proved that projective modules (over local rings) are free. The local assumption is really important: there are a lot of interesting research papers (even books) on the freeness of projective modules over polynomial rings with coefficients from a field. In general, the class of reflexive modules is extremely big compared to the projective modules. As a generalization of Seshadri’s result, Serre observed over 2-dimensional regular local rings that finitely generated reflexive modules are free in 1958. This result has some applications: For instance, in the arithmetical property of Iwasawa algebras (see [45]). It seems freeness of reflexive modules is subtle even over very special rings. For example, in = k[X,Y] [29, Page 518] Lam says that the only obvious examples of reflexive modules over R : (X,Y)2 are the free modules Rn. Ramras posed the following: Problem 1.1. (See [19, Page 380]) When are finitely generated reflexive modules free? Over quasi-reduced rings, problem 1.1 was completely answered (see Proposition 4.22). -
The Depth Theory of Hopf Algebras and Smash Products
The Depth Theory of Hopf Algebras and Smash Products Christopher J. Young Supervised by Dr. Lars Kadison November 25, 2016 To all my loved ones. Abstract The work done during this doctoral thesis involved advancing the theory of algebraic depth. Subfactor depth is a concept which had already existed for decades, but research papers discovered a purely algebraic analogue to the concept. The main uses of algebraic depth, which applies to a ring and subring pair, have been in Hopf-Galois theory, Hopf algebra actions and to some extent group theory. In this thesis we consider the application of algebraic depth to finite dimensional Hopf algebras, and smash products. This eventually leads to a striking discovery, a concept of module depth. Before explaining this work the thesis will go through many know results of depth up to this point historically. The most important result of the thesis: we discover a strong connection between the algebraic depth of a smash product A#H and the module depth of an H-module algebra A. In separate work L. Kadison discovered a connection between algebraic depth R ⊆ H for Hopf algebras and the module depth of V ∗, another important H-module algebra. The three concepts are related. Another important achievement of the work herein, we are able to calculate for the first time depth values of polynomial algebras, Taft algebras with certain subgroups and specific smash products of the Taft algebras. We also give a bound for the depth of R=I ⊆ H=I, which is extremely useful in general. Contents Background iii Contribution iii 1 The Depth Theory of a Ring Extension 1 1.1 Tensor Products . -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
![Arxiv:1502.01692V2 [Math.DG]](https://docslib.b-cdn.net/cover/8359/arxiv-1502-01692v2-math-dg-1078359.webp)
Arxiv:1502.01692V2 [Math.DG]
LIMITING CONFIGURATIONS FOR SOLUTIONS OF HITCHIN’S EQUATION RAFE MAZZEO, JAN SWOBODA, HARTMUT WEISS, AND FREDERIK WITT Abstract. We review recent work on the compactification of the mod- uli space of Hitchin’s self-duality equation. We study the degeneration behavior near the ends of this moduli space in a set of generic directions by showing how limiting configurations can be desingularized. Following ideas of Hitchin, we can relate the top boundary stratum of this space of limiting configurations to a Prym variety. A key rˆole is played by the family of rotationally symmetric solutions to the self-duality equation on C, which we discuss in detail here. Contents 1. Introduction 1 2. Holomorphic bundles with Higgs fields 3 2.1. Stable bundles 3 2.2. Higgs bundles 5 2.3. Parabolic Higgs bundles 7 2.4. Spectral curves 8 3. Limiting configurations 9 3.1. Motivation 9 3.2. The fiducial solution 9 3.3. Construction of limiting configurations 13 4. Desingularization by gluing 16 5. The hyperk¨ahler metric 19 5.1. Moment maps 19 arXiv:1502.01692v2 [math.DG] 2 Jul 2015 5.2. The semi-flat metric 20 References 21 1. Introduction The moduli space of Higgs bundles, introduced by Hitchin [Hi87] and Simpson [Si88], is a well investigated object in algebraic geometry and topol- ogy. We wish here to study it from the viewpoint of Riemannian geometry. Hitchin showed that there exists a natural hyperk¨ahler metric on the smooth Date: July 17, 2021. RM supported by NSF Grant DMS-1105050, JS supported by DFG Grant Sw 161/1-1. -
The Eagle 2005
CONTENTS Message from the Master .. .. .... .. .... .. .. .. .. .. .... ..................... 5 Commemoration of Benefactors .. .............. ..... ..... ....... .. 10 Crimes and Punishments . ................................................ 17 'Gone to the Wars' .............................................. 21 The Ex-Service Generations ......................... ... ................... 27 Alexandrian Pilgrimage . .. .. .. .. .. .. .. .. .. .. .. .................. 30 A Johnian Caricaturist Among Icebergs .............................. 36 'Leaves with Frost' . .. .. .. .. .. .. ................ .. 42 'Chicago Dusk' .. .. ........ ....... ......... .. 43 New Court ........ .......... ....................................... .. 44 A Hidden Treasure in the College Library ............... .. 45 Haiku & Tanka ... 51 and sent free ...... 54 by St John's College, Cambridge, The Matterhorn . The Eagle is published annually and other interested parties. Articles members of St John's College .... 55 of charge to The Eagle, 'Teasel with Frost' ........... should be addressed to: The Editor, to be considered for publication CB2 1 TP. .. .. .... .. .. ... .. ... .. .. ... .... .. .. .. ... .. .. 56 St John's College, Cambridge, Trimmings Summertime in the Winter Mountains .. .. ... .. .. ... ... .... .. .. 62 St John's College Cambridge The Johnian Office ........... ..... .................... ........... ........... 68 CB2 1TP Book Reviews ........................... ..................................... 74 http:/ /www.joh.cam.ac.uk/ Obituaries -
Differential Geometry and the Quaternions
DIFFERENTIAL GEOMETRY AND THE QUATERNIONS Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013 3 26th December 1843 • 16th October 1843 2 3 RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 965 We call a frame an ordered set of four mutually perpendicular unit vectors eo, Ci, C2, 63. There exists one and only one rotation carrying 3 one frame to another. The coordinates Xof X\, &2, Xz of a point ï&S with respect to the frame eo, ei, C2, e3 are defined by the equation (2) % = XQCO + Xtfi + X2t2 + Xst3. Let eo*, ei*, e2*, e3* be a frame related to eo, ei, e2, e3 by means of the rela- tions 3 (3) e«* = ]F) daftp, a = 0, 1, 2, 3, OwherN RIEMANNIAe (aap) is a propeN MANIFOLDr orthogonaSl OmatrixF FOU, anR d DIMENSIONSlet #0*, #1*, #2*1 , #3* be the coordinates of the same point x with respect to the frame SHIING-SHEN CHERN eo*, ei*, e2*, e3*. Then we have Introduction. It is well known that3 in three-dimensional elliptic or spherica(3a) l geometry the so-callex*d Clifford'= X) a<*p%p,s parallelis m or parataxot = 0,y 1ha, 2s, 3. locallymany interestinµ = µ ωg properties+ µ ω .+ Aµ group-theoretiω c reason for the most • 1 1 2 2 3 3 importanThe propertiet of thess oef propertiespherical sgeometr is the facy art etha thost the ewhich universa, whel ncoverin expresseg d grouin pterm of ths eo fprope coordinater orthogonas withl grourespecp itn tfouo ar framevariable, remais is thn e invariandirect t ifproducµunde= 0tr chango distinguishedf thee universaof the framel almostcoverin. -
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. -
Cohen-Macaulay Rings and Schemes
Cohen-Macaulay rings and schemes Caleb Ji Summer 2021 Several of my friends and I were traumatized by Cohen-Macaulay rings in our commuta- tive algebra class. In particular, we did not understand the motivation for the definition, nor what it implied geometrically. The purpose of this paper is to show that the Cohen-Macaulay condition is indeed a fruitful notion in algebraic geometry. First we explain the basic defini- tions from commutative algebra. Then we give various geometric interpretations of Cohen- Macaulay rings. Finally we touch on some other areas where the Cohen-Macaulay condition shows up: Serre duality and the Upper Bound Theorem. Contents 1 Definitions and first examples1 1.1 Preliminary notions..................................1 1.2 Depth and Cohen-Macaulay rings...........................3 2 Geometric properties3 2.1 Complete intersections and smoothness.......................3 2.2 Catenary and equidimensional rings.........................4 2.3 The unmixedness theorem and miracle flatness...................5 3 Other applications5 3.1 Serre duality......................................5 3.2 The Upper Bound Theorem (combinatorics!)....................6 References 7 1 Definitions and first examples We begin by listing some relevant foundational results (without commentary, but with a few hints on proofs) of commutative algebra. Then we define depth and Cohen-Macaulay rings and present some basic properties and examples. Most of this section and the next are based on the exposition in [1]. 1.1 Preliminary notions Full details regarding the following standard facts can be found in most commutative algebra textbooks, e.g. Theorem 1.1 (Nakayama’s lemma). Let (A; m) be a local ring and let M be a finitely generated A-module.