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The William Lowell Putnam Mathematical Competition 1985–2000 Problems, Solutions, and Commentary
The William Lowell Putnam Mathematical Competition 1985–2000 Problems, Solutions, and Commentary i Reproduction. The work may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents. In any reproduction, the original publication by the Publisher must be credited in the following manner: “First published in The William Lowell Putnam Mathematical Competition 1985–2000: Problems, Solutions, and Commen- tary, c 2002 by the Mathematical Association of America,” and the copyright notice in proper form must be placed on all copies. Ravi Vakil’s photo on p. 337 is courtesy of Gabrielle Vogel. c 2002 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2002107972 ISBN 0-88385-807-X Printed in the United States of America Current Printing (last digit): 10987654321 ii The William Lowell Putnam Mathematical Competition 1985–2000 Problems, Solutions, and Commentary Kiran S. Kedlaya University of California, Berkeley Bjorn Poonen University of California, Berkeley Ravi Vakil Stanford University Published and distributed by The Mathematical Association of America iii MAA PROBLEM BOOKS SERIES Problem Books is a series of the Mathematical Association of America consisting of collections of problems and solutions from annual mathematical competitions; compilations of problems (including unsolved problems) specific to particular branches of mathematics; books on the art and practice of problem solving, etc. Committee on Publications Gerald Alexanderson, Chair Roger Nelsen Editor Irl Bivens Clayton Dodge Richard Gibbs George Gilbert Art Grainger Gerald Heuer Elgin Johnston Kiran Kedlaya Loren Larson Margaret Robinson The Inquisitive Problem Solver, Paul Vaderlind, Richard K. -
CURRICULUM VITAE Name VASUDEVAN SRINIVAS Date And
CURRICULUM VITAE Name VASUDEVAN SRINIVAS Date and Place of Birth 6th June, 1958 at Delhi, India Citizenship Indian Present position Senior Professor, Tata Institute of Fundamental Research, Mumbai, India Mailing Address School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India. e-mail [email protected] Telephone (91)(22) 22782217 (Off.); (91)(22) 22804966 (Res.) Fax (91)(22) 22804666 OR 22804610 OR 22804611 Education B.Sc., 1977, St. Joseph’s College, Bangalore University, Bangalore, India M.S., 1978, and Ph.D., 1982, University of Chicago, Chicago, IL., USA. Research field Algebraic Geometry. Subfields of interest (i) Algebraic cycles (ii) Commutative Algebra (iii) Characteristic p methods (iv) Algebraic K-theory Visiting positions held IAS (Princeton), Duke University (Durham), Northeastern University (Boston), Uni- versity of Utah, University of Chicago, Math. Sciences Res. Institute (Berkeley), U. of Michigan (Ann Arbor), USA; Max Planck Inst. (Bonn), and University of Essen, Germany; Univ. Paris Sud (Orsay), and Univ. Paris VII, France; UNAM and UAM, Mexico. Awards and distinctions Indian National Science Academy Medal for Young Scientists, 1987; elected Fellow of Indian Academy of Sciences, 1994; B. M. Birla Science Award received in 1995; Awarded Swarnajayanthi Fellowship, 1998; Bhatnagar Prize, 2003; J.C.Bose Fellowship, 2008; TWAS Mathematics Prize, 2008; elected Fellow of INSA, 2008; invited speaker at ICM 2010; elected Member at Large (2011-2014), Executive Committee, International Mathematical Union. Editorial Board Memberships (i) Asian Journal of Mathematics (www.ims.cuhk.edu.hk/~ajm/) (ii) Algebra and Number Theory (www.jant.org). (iii) Proceedings of the International Congress of Mathematicians, Hyderabad, 2010. -
Abramo Hefez
We certify that Abramo Hefez --- Universidade Federal Fluminense participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““Singularities“Singularities of analytic plane curvescurves”””” Rio de Janeiro, July 31, 2008. Jacob Palis Organizing Committee We certify that Adimurthi --- Tata Institute of Fundamental Research participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““Hardy“Hardy Soblev Rellich InequalitiesInequalities”””” Rio de Janeiro, July 31, 2008. Jacob Palis Organizing Committee We certify that Antar BandyopadBandyopadhyayhyay --- Indian Statistical Institute, Delhi Centre participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““Annealed“Annealed and Quenched IP for Random Walk in Dynamic Markovian EnvironmentEnvironment”””” Rio de Janeiro, July 31, 2008. Jacob Palis Organizing Committee We certify that Antonio Roberto da Silva --- Universidade Federal do Rio de Janeiro participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““Modified“Modified symplectic structures in cotangent bundlesbundles of Lie groupsgroups”””” Rio de Janeiro, July 31, 2008. Jacob Palis Organizing Committee We certify that Arnaldo Leite Pinto Garcia --- Instituto Nacional de Matemática Pura e Aplicada participated in the “First Indo-Brazilian Symposium in Mathematics”, held at IMPA, Rio de Janeiro from July 28 to August 01, 2008 and presented the following work: “““On“On curves and towers over finite fieldsfields”””” Rio de Janeiro, July 31, 2008. -
Mathematical Genealogy of the Wellesley College Department Of
Nilos Kabasilas Mathematical Genealogy of the Wellesley College Department of Mathematics Elissaeus Judaeus Demetrios Kydones The Mathematics Genealogy Project is a service of North Dakota State University and the American Mathematical Society. http://www.genealogy.math.ndsu.nodak.edu/ Georgios Plethon Gemistos Manuel Chrysoloras 1380, 1393 Basilios Bessarion 1436 Mystras Johannes Argyropoulos Guarino da Verona 1444 Università di Padova 1408 Cristoforo Landino Marsilio Ficino Vittorino da Feltre 1462 Università di Firenze 1416 Università di Padova Angelo Poliziano Theodoros Gazes Ognibene (Omnibonus Leonicenus) Bonisoli da Lonigo 1477 Università di Firenze 1433 Constantinople / Università di Mantova Università di Mantova Leo Outers Moses Perez Scipione Fortiguerra Demetrios Chalcocondyles Jacob ben Jehiel Loans Thomas à Kempis Rudolf Agricola Alessandro Sermoneta Gaetano da Thiene Heinrich von Langenstein 1485 Université Catholique de Louvain 1493 Università di Firenze 1452 Mystras / Accademia Romana 1478 Università degli Studi di Ferrara 1363, 1375 Université de Paris Maarten (Martinus Dorpius) van Dorp Girolamo (Hieronymus Aleander) Aleandro François Dubois Jean Tagault Janus Lascaris Matthaeus Adrianus Pelope Johann (Johannes Kapnion) Reuchlin Jan Standonck Alexander Hegius Pietro Roccabonella Nicoletto Vernia Johannes von Gmunden 1504, 1515 Université Catholique de Louvain 1499, 1508 Università di Padova 1516 Université de Paris 1472 Università di Padova 1477, 1481 Universität Basel / Université de Poitiers 1474, 1490 Collège Sainte-Barbe -
FOCUS August/September 2005
FOCUS August/September 2005 FOCUS is published by the Mathematical Association of America in January, February, March, April, May/June, FOCUS August/September, October, November, and Volume 25 Issue 6 December. Editor: Fernando Gouvêa, Colby College; [email protected] Inside Managing Editor: Carol Baxter, MAA 4 Saunders Mac Lane, 1909-2005 [email protected] By John MacDonald Senior Writer: Harry Waldman, MAA [email protected] 5 Encountering Saunders Mac Lane By David Eisenbud Please address advertising inquiries to: Rebecca Hall [email protected] 8George B. Dantzig 1914–2005 President: Carl C. Cowen By Don Albers First Vice-President: Barbara T. Faires, 11 Convergence: Mathematics, History, and Teaching Second Vice-President: Jean Bee Chan, An Invitation and Call for Papers Secretary: Martha J. Siegel, Associate By Victor Katz Secretary: James J. Tattersall, Treasurer: John W. Kenelly 12 What I Learned From…Project NExT By Dave Perkins Executive Director: Tina H. Straley 14 The Preparation of Mathematics Teachers: A British View Part II Associate Executive Director and Director By Peter Ruane of Publications: Donald J. Albers FOCUS Editorial Board: Rob Bradley; J. 18 So You Want to be a Teacher Kevin Colligan; Sharon Cutler Ross; Joe By Jacqueline Brennon Giles Gallian; Jackie Giles; Maeve McCarthy; Colm 19 U.S.A. Mathematical Olympiad Winners Honored Mulcahy; Peter Renz; Annie Selden; Hortensia Soto-Johnson; Ravi Vakil. 20 Math Youth Days at the Ballpark Letters to the editor should be addressed to By Gene Abrams Fernando Gouvêa, Colby College, Dept. of 22 The Fundamental Theorem of ________________ Mathematics, Waterville, ME 04901, or by email to [email protected]. -
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 -
THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION 1985–2000 Problems, Solutions, and Commentary
AMS / MAA PROBLEM BOOKS VOL 33 THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION 1985–2000 Problems, Solutions, and Commentary Kiran S. Kedlaya Bjorn Poonen Ravi Vakil 10.1090/prb/033 The William Lowell Putnam Mathematical Competition 1985-2000 Originally published by The Mathematical Association of America, 2002. ISBN: 978-1-4704-5124-0 LCCN: 2002107972 Copyright © 2002, held by the American Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2019 The American Mathematical Society retains all rights except those granted to the United States Government. ⃝1 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 24 23 22 21 20 19 AMS/MAA PROBLEM BOOKS VOL 33 The William Lowell Putnam Mathematical Competition 1985-2000 Problems, Solutions, and Commentary Kiran S. Kedlaya Bjorn Poonen Ravi Vakil MAA PROBLEM BOOKS SERIES Problem Books is a series of the Mathematical Association of America consisting of collections of problems and solutions from annual mathematical competitions; compilations of problems (including unsolved problems) specific to particular branches of mathematics; books on the art and practice of problem solving, etc. Committee on Publications Gerald Alexanderson, Chair Problem Books Series Editorial Board Roger Nelsen Editor Irl Bivens Clayton Dodge Richard Gibbs George Gilbert Art Grainger Gerald Heuer Elgin Johnston Kiran Kedlaya Loren Larson Margaret Robinson The Contest Problem Book VII: American Mathematics Competitions, 1995-2000 Contests, compiled and augmented by Harold B. -
MAA Awards Given in Baltimore
MAA Awards Given in Baltimore At the Joint Mathematics Meetings in Baltimore, concluded that “highly integrated content-methods Maryland, in January 2014, the Mathematical As- instruction provided no measurable advantage sociation of America awarded several prizes. for students but that coordination of separate courses in content and methods served to improve Gung and Hu Award for Distinguished students’ performance in both.” Another step was Service the early placement testing for high school juniors, The Yueh-Gin Gung and Dr. Charles Y. Hu Award followed by an appropriate twelfth-grade course for Distinguished Service to Mathematics is the to prepare for college mathematics. One problem most prestigious award made by the MAA. It revealed by early placement testing was that many honors distinguished contributions to mathemat- high school juniors had very limited algebra skills ics and mathematical education in one particular and no appropriate course for them to take in aspect or many, whether in a short period or over their senior year. With funding from the Battelle a career. Foundation, Leitzel and Demana undertook the The 2014 Gung and Hu Award was presented to project A Numerical Problem-Solving Course for Joan Leitzel for her farsighted work of creating Underprepared College-Intending Seniors. The programs to decrease the need for remediation course they developed took a highly numerical in colleges and for her leadership on the national approach to algebra and geometry and very suc- level. Her distinguished career began at Ohio State cessfully brought students up to competency at the University, where she led visionary projects that level of intermediate algebra. -
2004 Notices Index
The 2004 Notices Index Index Section Page Number Index Section Page Number The AMS 1413 Memorial Articles 1424 Announcements 1414 New Publications Offered by the AMS 1425 Authors of Articles 1416 Officers and Committee Members 1425 Deaths of Members of the Society 1417 Opinion 1425 Education 1418 Opportunities 1425 Feature Articles 1418 Prizes and Awards 1426 Letters to the Editor 1419 The Profession 1428 Mathematicians 1420 Reference and Book List 1428 Mathematics Articles 1423 Reviews 1428 Mathematics History 1424 Surveys 1428 Meetings Information 1424 Tables of Contents 1428 About the Cover AMS-AAAS Mass Media Summer Fellowships, 1075 8, 207, 319, 411, 495, 619, 815, 883, 1023, 1194, 1347 AMS-AAAS Media Fellow Named, 808 AMS Associate Executive Editor Positions—Applications Invited, 602 AMS Book Prize, 450, 568 AMS Committee on Meetings and Conferences, 1249 AMS Committee on Science Policy, 1246 AMS Email Support for Frequently Asked Questions, 1365 AMS Menger Awards at the 2004 ISEF, 1235 AMS Menger Prizes at the 2004 ISEF, 913 The AMS AMS Mentoring Workshop, 1304 2000 Mathematics Subject Classification, 60, 275, 454, AMS Officers and Committee Members, 1082 572, 992, 1290 AMS Participates in Capitol Hill Exhibition, 1076 2003 AMS Election Results, 269 AMS Participates in Celebration of von Neumann’s Birth, 2003 AMS Policy Committee Reports, 264 52 2003 Annual Survey of the Mathematical Sciences (First AMS Scholarships for “Math in Moscow”, 805 Report), 218 AMS Short Course, 1161 2003 Annual Survey of the Mathematical Sciences (Second -
The 1999 Notices Index, Volume 46, Number 11
index.qxp 10/15/99 11:17 AM Page 1480 The 1999 Notices Index Index Section Page Number The AMS 1480 Meetings Information 1489 Announcements 1481 Memorial Articles 1490 Authors of Articles 1482 New Publications Offered by the AMS 1490 Reference and Book List 1420 Officers and Committee Members 1490 Deaths of Members of the Society 1483 Opinion 1490 Education 1484 Opportunities 1490 Feature Articles 1484 Prizes and Awards 1491 Forum 1485 Articles on the Profession 1492 Letters to the Editor 1486 Reviews 1492 Mathematicians 1487 Surveys 1493 Mathematics Articles 1489 Tables of Contents 1493 Mathematics History 1489 About the Cover, 26, 245, 318, 420, 534, 658, 781, 918, AMS Meeting in August 2000, 355 1073, 1208, 1367 AMS Menger Awards at the 1999 Intel-International Science and Engineering Fair, 912 Acknowledgment of Contributions, 947 AMS Officers and Committee Members, 1271 AMS Participates in Capitol Hill Exhibition, 918 The AMS AMS Participates in Faculty Preparation Program, 1072 1998 Election Results, 266 AMS Short Courses, 1171 1998 Report of the Committee on Meetings and Confer- AMS Standard Cover Sheet, 60, 270, 366, 698, 972, 1102, ences, 810 1282, 1428 1999 AMS Centennial Fellowships Awarded, 684 (The) Birth of an AMS Meeting, 1029 1999 AMS Election, 58, 267 Bylaws of the American Mathematical Society, 1252 1999 AMS-MAA-SIAM Morgan Prize, 469 Call for Proposals for 2001 Conferences, 1177, 1331 1999 Arnold Ross Lectures, 692 Conference Announced (Summer Mathematics Experience: 1999 Bôcher Prize, 463 A Working Conference on Summer -
Profiles and Prospects*
Indian Journal of History of Science, 47.3 (2012) 473-512 MATHEMATICS AND MATHEMATICAL RESEARCHES IN INDIA DURING FIFTH TO TWENTIETH CENTURIES — PROFILES AND PROSPECTS* A K BAG** (Received 1 September 2012) The Birth Centenary Celebration of Professor M. C. Chaki (1912- 2007), former First Asutosh Birth Centenary Professor of Higher Mathematics and a noted figure in the community of modern geometers, took place recently on 21 July 2012 in Kolkata. The year 2012 is also the 125th Birth Anniversary Year of great mathematical prodigy, Srinivas Ramanujan (1887-1920), and the Government of India has declared 2012 as the Year of Mathematics. To mark the occasion, Dr. A. K. Bag, FASc., one of the students of Professor Chaki was invited to deliver the Key Note Address. The present document made the basis of his address. Key words: Algebra, Analysis, Binomial expansion, Calculus, Differential equation, Fluid and solid mechanics, Function, ISI, Kerala Mathematics, Kut..taka, Mathemtical Modeling, Mathematical Societies - Calcutta, Madras and Allahabad, Numbers, Probability and Statistics, TIFR, Universities of Calcutta, Madras and Bombay, Vargaprakr.ti. India has been having a long tradition of mathematics. The contributions of Vedic and Jain mathematics are equally interesting. However, our discussion starts from 5th century onwards, so the important features of Indian mathematics are presented here in phases to make it simple. 500-1200 The period: 500-1200 is extremely interesting in the sense that this is known as the Golden (Siddha–ntic) period of Indian mathematics. It begins * The Key Note Address was delivered at the Indian Association for Cultivation of Science (Central Hall of IACS, Kolkata) organized on behalf of the M.C. -
Curriculum Vitae: Brendan Hassett
Curriculum Vitae: Brendan Hassett Brown University brendan [email protected] Institute for Computational and Experimental Research in Mathematics 121 South Main Street, 11th Floor, Box E Providence, RI 02903 USA 401 863 7010 Department of Mathematics Box 1917 151 Thayer Street Providence, Rhode Island 02912 USA 401 863 7961 Positions Director, Institute for Computational and Experimental Research in Mathemat- ics, Brown University, from July 2016 Professor of Mathematics, from July 2015 Milton Brockett Porter Professor, Rice University, July 2013 to June 2015 Chair, Department of Mathematics, July 2009 to June 2014 Professor of Mathematics, July 2006 to June 2015 Associate Professor of Mathematics, July 2003 to June 2006 Assistant Professor of Mathematics, July 2000 to June 2003 Professeur Invit´e,Universit´eParis-Sud, Orsay, March to April 2005 Visiting Scholar, Institute of Mathematical Sciences, Chinese University of Hong Kong, August 2000 to July 2001 Dickson Instructor of Mathematics, University of Chicago, October 1996 to September 2000 Visitor at the Institut Mittag-Leffler, Stockholm, January to March 1997 Education Harvard University, M.A., 1994, and Ph.D., 1996 (supervised by Joe Harris) Yale College, B.A. in mathematics, summa cum laude, 1992 Awards Fellow of the American Mathematical Society, 2014 Charles W. Duncan Jr. Achievement Award for Outstanding Faculty, Rice Uni- versity, 2009 Grants and Fellowships Simons Foundation Award 815891: September 1, 2021{August 31, 2023; Simons Bridge for Postdoctoral Fellowships at ICERM