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Field Theory Pete L. Clark
Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9. -
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 -
Cédric VILLANI: Curriculum Vitae (Last Updated August 4, 2012)
C´edric VILLANI: Curriculum Vitae (last updated August 4, 2012) Professor of the Universit´ede Lyon Director of the Institut Henri Poincar´e 11 rue Pierre et Marie Curie, F-75230 Paris Cedex 05, FRANCE. Tel.: +33 1 44 27 64 18, fax: +33 1 46 34 04 56. E-Mail: [email protected] Internet address: http://math.univ-lyon1.fr/~villani Personal information - Born October 5, 1973 in Brive-la-Gaillarde (France); french citizen - age 38, two children - languages: french (native), english (fluent), italian - hobbies: walking, music (piano) Positions held - From 2000 to 2010 I have been professor (mathematics) in the Ecole´ Normale Sup´erieure de Lyon, where I did research and teaching up to graduate level. In September 2010 I moved to the Universit´eClaude Bernard Lyon I. - Since July 2009 I am director of the Institut Henri Poincar´e(Paris), where I do research and administration. I am the coordinator of the CARMIN structure, which gathers the four international french institutes for mathematics: CIRM, CIMPA, IHP, IHES.´ - Invited member of the Institute for Advanced Study, Princeton (January–June 2009) - Visiting Research Miller Professor at the University of Berkeley (January–May 2004) - Visiting Assistant Professor at the Georgia Tech Institute, Atlanta (Fall 1999) - Student, then agr´eg´e-pr´eparateur (“tutor”) at the ENS, Paris (1992-2000) Diplomas, titles and awards - Fields Medal (2010) - Fermat Prize (2009) - Henri Poincar´ePrize of the International Association of Mathematical Physics (2009) - Prize of the European Mathematical Society (2008) - Jacques Herbrand Prize of the Academy of Sciences (2007) - Invited lecturer at the International Congress of Mathematicians (Madrid, 2006) - Institut Universitaire de France (2006) - Harold Grad lecturer (2004) - Plenary lecturer at the International Congress of Mathematical Physics (Lisbonne, 2003) - Peccot-Vimont Prize and Cours Peccot of the Coll`ege de France (2003) - Louis Armand Prize of the Academy of Sciences (2001) - PhD Thesis (1998; advisor P.-L. -
For Your Information
fyi.qxp 3/18/98 3:19 PM Page 244 For Your Information structure federal support. The study is part of a larger ef- International Study of fort at the Academy to gauge where the U.S. stands in- Mathematics and Science ternationally in scientific research. The motivation comes from a 1993 report by the Committee on Science, Engi- Achievement neering, and Public Policy (COSEPUP), which set forth strate- gies for making decisions about how best to use federal The first installment of results from the Third International research funds. COSEPUP is a committee of the NAS, the Mathematics and Science Study (TIMSS) was released on No- National Academy of Engineering, and the Institute of Med- vember 20. This first batch of data pertains to achievement icine. The COSEPUP report recommended that the U.S. aim of eighth-graders; later releases will focus on fourth- and to be the world leader in certain critical fields and to be twelfth-graders. The study found that U.S. eighth-graders among the leaders in other areas. The report urged field- performed below the international average in mathemat- by-field assessments by independent panels of researchers ics but slightly above average in science. The U.S. was in the field, researchers in closely related fields, and users among thirty-three countries in which there was no sta- of the research. The mathematical sciences study is the first tistically significant difference between the performance such assessment. If this project is successful, the Academy of eighth-grade boys and girls in mathematics. The study will follow suit with other areas. -
Geometry on Grassmannians and Applications to Splitting Bundles and Smoothing Cycles
PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. STEVEN L. KLEIMAN Geometry on grassmannians and applications to splitting bundles and smoothing cycles Publications mathématiques de l’I.H.É.S., tome 36 (1969), p. 281-297. <http://www.numdam.org/item?id=PMIHES_1969__36__281_0> © Publications mathématiques de l’I.H.É.S., 1969, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ GEOMETRY ON GRASSMANNIANS AND APPLICATIONS TO SPLITTING BUNDLES AND SMOOTHING CYCLES by STEVEN L. KLEIMAN (1) INTRODUCTION Let k be an algebraically closed field, V a smooth ^-dimensional subscheme of projective space over A:, and consider the following problems: Problem 1 (splitting bundles). — Given a (vector) bundle G on V, find a monoidal transformation f \ V'->V with smooth center CT such thatyG contains a line bundle P. Problem 2 (smoothing cycles). — Given a cycle Z on V, deform Z by rational equivalence into the difference Z^ — Zg of two effective cycles whose prime components are all smooth. Strengthened form. — Given any finite number of irreducible subschemes V, of V, choose a (resp. Z^, Zg) such that for all i, the intersection V^n a (resp. -
Symplectic Embeddings of Toric Domains
Symplectic Embeddings of Toric Domains by Daniel Reid Irvine A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University of Michigan 2020 Doctoral Committee: Professor Daniel Burns, Jr., Chair Professor Anthony Bloch Assistant Professor Jun Li Professor John Schotland Professor Alejandro Uribe Daniel Reid Irvine [email protected] ORCID ID: 0000-0003-2721-5901 ©Daniel Reid Irvine 2020 Dedication This work is gratefully dedicated to my family. ii Acknowledgments I would like to extend my deep gratitude to my advisor, Daniel Burns, for his continued support and encouragement during my graduate career. Beyond his mathematical exper- tise, he has been a steadfast friend. Second, I would like to thank Richard Hind for his advise- ment during both my undergraduate and graduate career. He encouraged me to undertake this ambitious project, and his enthusiasm steered me towards the field of symplectic ge- ometry. Third, I would like to thank Jun Li for his advice and many helpful conversations. Finally, I would like to thank all the instructors that guided me through my mathematical career. I would not be where I am today if not for them. iii Table of Contents Dedication ....................................... ii Acknowledgments ................................... iii List of Figures ..................................... vi Abstract ......................................... vii Chapter 1 Introduction ..................................... 1 1.0.1 Summary of Results........................6 1.0.2 Important Past Results.......................7 2 Toric Domains .................................... 11 2.1 ECH Capacities of Toric Domains..................... 13 2.1.1 ECH Capacities of Concave Toric Domains............ 14 2.1.2 Realizing the positive weight vector as a ball packing...... -
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Volume 47 • Issue 1 IMS Bulletin January/February 2018 National Academy new member The US National Academy of Medicine (NAM) has announced the election of 70 reg- CONTENTS ular members and 10 international members. Among them is Nicholas Patrick Jewell, 1 National Academy of University of California, Berkeley. Medicine elects Jewell Election to the Academy is considered one of the highest honors in the fields of 2 Members’ news: Nick Horton, health and medicine, recognizing individuals who have made major contributions to Eric Kolaczyk, Hongzhe Li, the advancement of the medical sciences, health care, and public health. A diversity Runze Li, Douglas Simpson, of talent among NAM’s membership is Greg Lawler, Mike Jordan, Mir assured by its Articles of Organization, Masoom Ali which stipulate that at least one-quarter 3 Journal news: EJP, ECP, Prob of the membership is selected from fields Surveys; OECD guidelines outside the health professions, for exam- 4 Takis Konstantopoulos: new ple, from law, engineering, social sciences, column and the humanities—and statistics. The newly elected members bring 6 Recent papers: Electronic Journal of Probability; Electronic NAM’s total membership to 2,127 and Communications in Probability the number of international members to 172. 11 Special Invited Lecturers; IMS Fellow Nicholas P. Jewell is New Textbook Professor of Biostatistics and Statistics 12 Obituary: Ron Getoor at the University of California, Berkeley. 13 IMS Awards Since arriving at Berkeley in 1981, he has held various academic and administrative 15 Student Puzzle Corner 19; New Researcher Award positions, most notably serving as Vice Provost from 1994 to 2000. -
Comparative Literature in Slovenia
CLCWeb: Comparative Literature and Culture ISSN 1481-4374 Purdue University Press ©Purdue University Volume 2 (2000) Issue 4 Article 11 Comparative Literature in Slovenia Kristof Jacek Kozak University of Alberta Follow this and additional works at: https://docs.lib.purdue.edu/clcweb Part of the Comparative Literature Commons, and the Critical and Cultural Studies Commons Dedicated to the dissemination of scholarly and professional information, Purdue University Press selects, develops, and distributes quality resources in several key subject areas for which its parent university is famous, including business, technology, health, veterinary medicine, and other selected disciplines in the humanities and sciences. CLCWeb: Comparative Literature and Culture, the peer-reviewed, full-text, and open-access learned journal in the humanities and social sciences, publishes new scholarship following tenets of the discipline of comparative literature and the field of cultural studies designated as "comparative cultural studies." Publications in the journal are indexed in the Annual Bibliography of English Language and Literature (Chadwyck-Healey), the Arts and Humanities Citation Index (Thomson Reuters ISI), the Humanities Index (Wilson), Humanities International Complete (EBSCO), the International Bibliography of the Modern Language Association of America, and Scopus (Elsevier). The journal is affiliated with the Purdue University Press monograph series of Books in Comparative Cultural Studies. Contact: <[email protected]> Recommended Citation Kozak, Kristof Jacek. "Comparative Literature in Slovenia." CLCWeb: Comparative Literature and Culture 2.4 (2000): <https://doi.org/10.7771/1481-4374.1094> This text has been double-blind peer reviewed by 2+1 experts in the field. The above text, published by Purdue University Press ©Purdue University, has been downloaded 2344 times as of 11/ 07/19. -
Dynamics, Equations and Applications Book of Abstracts Session
DYNAMICS, EQUATIONS AND APPLICATIONS BOOK OF ABSTRACTS SESSION D21 AGH University of Science and Technology Kraków, Poland 1620 September 2019 2 Dynamics, Equations and Applications CONTENTS Plenary lectures 7 Artur Avila, GENERIC CONSERVATIVE DYNAMICS . .7 Alessio Figalli, ON THE REGULARITY OF STABLE SOLUTIONS TO SEMI- LINEAR ELLIPTIC PDES . .7 Martin Hairer, RANDOM LOOPS . .8 Stanislav Smirnov, 2D PERCOLATION REVISITED . .8 Shing-Tung Yau, STABILITY AND NONLINEAR PDES IN MIRROR SYMMETRY8 Maciej Zworski, FROM CLASSICAL TO QUANTUM AND BACK . .9 Public lecture 11 Alessio Figalli, FROM OPTIMAL TRANSPORT TO SOAP BUBBLES AND CLOUDS: A PERSONAL JOURNEY . 11 Invited talks of part D2 13 Stefano Bianchini, DIFFERENTIABILITY OF THE FLOW FOR BV VECTOR FIELDS . 13 Yoshikazu Giga, ON THE LARGE TIME BEHAVIOR OF SOLUTIONS TO BIRTH AND SPREAD TYPE EQUATIONS . 14 David Jerison, THE TWO HYPERPLANE CONJECTURE . 14 3 4 Dynamics, Equations and Applications Sergiu Klainerman, ON THE NONLINEAR STABILITY OF BLACK HOLES . 15 Aleksandr Logunov, ZERO SETS OF LAPLACE EIGENFUCNTIONS . 16 Felix Otto, EFFECTIVE BEHAVIOR OF RANDOM MEDIA . 17 Endre Süli, IMPLICITLY CONSTITUTED FLUID FLOW MODELS: ANALYSIS AND APPROXIMATION . 17 András Vasy, GLOBAL ANALYSIS VIA MICROLOCAL TOOLS: FREDHOLM THEORY IN NON-ELLIPTIC SETTINGS . 19 Luis Vega, THE VORTEX FILAMENT EQUATION, THE TALBOT EFFECT, AND NON-CIRCULAR JETS . 20 Enrique Zuazua, POPULATION DYNAMICS AND CONTROL . 20 Talks of session D21 23 Giovanni Bellettini, ON THE RELAXED AREA OF THE GRAPH OF NONS- MOOTH MAPS FROM THE PLANE TO THE PLANE . 23 Sun-Sig Byun, GLOBAL GRADIENT ESTIMATES FOR NONLINEAR ELLIP- TIC PROBLEMS WITH NONSTANDARD GROWTH . 24 Juan Calvo, A BRIEF PERSPECTIVE ON TEMPERED DIFFUSION EQUATIONS 25 Giacomo Canevari, THE SET OF TOPOLOGICAL SINGULARITIES OF VECTOR- VALUED MAPS . -
Institute for Pure and Applied Mathematics, UCLA Award/Institution #0439872-013151000 Annual Progress Report for 2009-2010 August 1, 2011
Institute for Pure and Applied Mathematics, UCLA Award/Institution #0439872-013151000 Annual Progress Report for 2009-2010 August 1, 2011 TABLE OF CONTENTS EXECUTIVE SUMMARY 2 A. PARTICIPANT LIST 3 B. FINANCIAL SUPPORT LIST 4 C. INCOME AND EXPENDITURE REPORT 4 D. POSTDOCTORAL PLACEMENT LIST 5 E. INSTITUTE DIRECTORS‘ MEETING REPORT 6 F. PARTICIPANT SUMMARY 12 G. POSTDOCTORAL PROGRAM SUMMARY 13 H. GRADUATE STUDENT PROGRAM SUMMARY 14 I. UNDERGRADUATE STUDENT PROGRAM SUMMARY 15 J. PROGRAM DESCRIPTION 15 K. PROGRAM CONSULTANT LIST 38 L. PUBLICATIONS LIST 50 M. INDUSTRIAL AND GOVERNMENTAL INVOLVEMENT 51 N. EXTERNAL SUPPORT 52 O. COMMITTEE MEMBERSHIP 53 P. CONTINUING IMPACT OF PAST IPAM PROGRAMS 54 APPENDIX 1: PUBLICATIONS (SELF-REPORTED) 2009-2010 58 Institute for Pure and Applied Mathematics, UCLA Award/Institution #0439872-013151000 Annual Progress Report for 2009-2010 August 1, 2011 EXECUTIVE SUMMARY Highlights of IPAM‘s accomplishments and activities of the fiscal year 2009-2010 include: IPAM held two long programs during 2009-2010: o Combinatorics (fall 2009) o Climate Modeling (spring 2010) IPAM‘s 2010 winter workshops continued the tradition of focusing on emerging topics where Mathematics plays an important role: o New Directions in Financial Mathematics o Metamaterials: Applications, Analysis and Modeling o Mathematical Problems, Models and Methods in Biomedical Imaging o Statistical and Learning-Theoretic Challenges in Data Privacy IPAM sponsored reunion conferences for four long programs: Optimal Transport, Random Shapes, Search Engines and Internet MRA IPAM sponsored three public lectures since August. Noga Alon presented ―The Combinatorics of Voting Paradoxes‖ on October 5, 2009. Pierre-Louis Lions presented ―On Mean Field Games‖ on January 5, 2010. -
EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics the Open University Milton Keynes MK7 6AA, UK E-Mail: [email protected]
CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY EDITOR-IN-CHIEF ROBIN WILSON Department of Pure Mathematics The Open University Milton Keynes MK7 6AA, UK e-mail: [email protected] ASSOCIATE EDITORS VASILE BERINDE Department of Mathematics, University of Baia Mare, Romania e-mail: [email protected] NEWSLETTER No. 47 KRZYSZTOF CIESIELSKI Mathematics Institute March 2003 Jagiellonian University Reymonta 4 EMS Agenda ................................................................................................. 2 30-059 Kraków, Poland e-mail: [email protected] Editorial by Sir John Kingman .................................................................... 3 STEEN MARKVORSEN Department of Mathematics Executive Committee Meeting ....................................................................... 4 Technical University of Denmark Building 303 Introducing the Committee ............................................................................ 7 DK-2800 Kgs. Lyngby, Denmark e-mail: [email protected] An Answer to the Growth of Mathematical Knowledge? ............................... 9 SPECIALIST EDITORS Interview with Vagn Lundsgaard Hansen .................................................. 15 INTERVIEWS Steen Markvorsen [address as above] Interview with D V Anosov .......................................................................... 20 SOCIETIES Krzysztof Ciesielski [address as above] Israel Mathematical Union ......................................................................... 25 EDUCATION Tony Gardiner -
Mathematics Education in the Balkan Societies up to the WWI
UDC 51(091)“19“ Teaching Innovations, 2014, Volume 27, Issue 3, pp. 46–57 Received: 28 October 2014 Accepted: 5 November 2014 Snezana Lawrence1, PhD Original Article Bath Spa University, School of Education, United Kingdom Mathematics Education in the Balkan Societies Up To the WWI Abstract: Whilst the world is indebted to the Greeks for their development of geometry and to Islamic mathematicians for their development of algebra, the history of violence and wars of the Balkan peninsula meant that neither heritages of these two great mathematical cultures survived into the 19th century. Th is paper is based on the research done for the history of mathematics in the Balkans and will be limited to the develop- ment of mathematical education in three Balkan societies: Greek, Ottoman, and Serbian, culminating in the early 20th century. It will try to explain how the three cultures of mathematics education were conceptualized, and how their development was infl uenced by the mathematical cultures of Western Europe. Th e systems of schools and universities, the fi rst professors of mathematics at the universities in the three countries, mathemati- cal syllabi, and some of the fi rst textbooks in mathematics will be mentioned. Key words: 19th century mathematics education, Balkan mathematics, Greek mathematics, Ottoman mathematics, Serbian mathematics. Introduction and their military apparatus. Th e second focal point will be the mathematics of two Orthodox popula- Th is paper focuses on the development of tions then under the Ottoman Empire: Greeks and Balkan mathematics education, which will be cov- Serbs. Greeks were the predominant Orthodox eth- ered through the history of three national schools nie, both in terms of their heritage and cultural in- of mathematics.