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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. -
TWAS Fellowships Worldwide
CDC Round Table, ICTP April 2016 With science and engineering, countries can address challenges in agriculture, climate, health TWAS’s and energy. guiding principles 2 Food security Challenges Water quality for a Energy security new era Biodiversity loss Infectious diseases Climate change 3 A Globally, 81 nations fall troubling into the category of S&T- gap lagging countries. 48 are classified as Least Developed Countries. 4 The role of TWAS The day-to-day work of TWAS is focused in two critical areas: •Improving research infrastructure •Building a corps of PhD scholars 5 TWAS Research Grants 2,202 grants awarded to individuals and research groups (1986-2015) 6 TWAS’ AIM: to train 1000 PhD students by 2017 Training PhD-level scientists: •Researchers and university-level educators •Future leaders for science policy, business and international cooperation Rapidly growing opportunities P BRAZIL A K I N D I CA I RI A S AF TH T SOU A N M KENYA EX ICO C H I MALAYSIA N A IRAN THAILAND TWAS Fellowships Worldwide NRF, South Africa - newly on board 650+ fellowships per year PhD fellowships +460 Postdoctoral fellowships +150 Visiting researchers/professors + 45 17 Programme Partners BRAZIL: CNPq - National Council MALAYSIA: UPM – Universiti for Scientific and Technological Putra Malaysia WorldwideDevelopment CHINA: CAS - Chinese Academy of KENYA: icipe – International Sciences Centre for Insect Physiology and Ecology INDIA: CSIR - Council of Scientific MEXICO: CONACYT– National & Industrial Research Council on Science and Technology PAKISTAN: CEMB – National INDIA: DBT - Department of Centre of Excellence in Molecular Biotechnology Biology PAKISTAN: ICCBS – International Centre for Chemical and INDIA: IACS - Indian Association Biological Sciences for the Cultivation of Science PAKISTAN: CIIT – COMSATS Institute of Information INDIA: S.N. -
Millennium Prize for the Poincaré
FOR IMMEDIATE RELEASE • March 18, 2010 Press contact: James Carlson: [email protected]; 617-852-7490 See also the Clay Mathematics Institute website: • The Poincaré conjecture and Dr. Perelmanʼs work: http://www.claymath.org/poincare • The Millennium Prizes: http://www.claymath.org/millennium/ • Full text: http://www.claymath.org/poincare/millenniumprize.pdf First Clay Mathematics Institute Millennium Prize Announced Today Prize for Resolution of the Poincaré Conjecture a Awarded to Dr. Grigoriy Perelman The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads: The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude. The award of the Millennium Prize to Dr. Perelman was made in accord with their governing rules: recommendation first by a Special Advisory Committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles), then by the CMI Scientific Advisory Board (James Carlson, Simon Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with final decision by the Board of Directors (Landon T. -
LMS – EPSRC Durham Symposium
LMS – EPSRC Durham Symposium Anthony Byrne Grants and Membership Administrator 12th July 2016, Durham The work of the LMS for mathematics The charitable aims of the Society: Funding the advancement of mathematical knowledge Encouraging mathematical research and collaboration ’, George Legendre Celebrating mathematical 30 Pieces achievements Publishing and disseminating mathematical knowledge Advancing and promoting mathematics The attendees of the Young Researchers in Mathematics Conference 2015, held at Oxford Historical Moments of the London Mathematical Society 1865 Foundation of LMS at University College London George Campbell De Morgan organised the first meeting, and his father, Augustus De Morgan became the 1st President 1865 First minute book list of the 27 original members 1866 LMS moves to Old Burlington House, Piccadilly J.J. Sylvester, 2nd President of the Society. 1866 Julius Plûcker Thomas Hirst Plûcker Collection of boxwood models of quartic surfaces given to Thomas Archer Hirst, Vice- President of LMS, and donated to the Society 1870 Move to Asiatic Society, 22 Albemarle Street William Spottiswoode, President 1874 Donation of £1,000 from John William Strutt (Lord Rayleigh) Generous donation enabled the Society to publish volumes of the Proceedings of the London Mathematical Society. J.W. Strutt (Lord Rayleigh), LMS President 1876-78 1881 First women members Charlotte Angas Scott and Christine Ladd 1884 First De Morgan medal awarded to Arthur Cayley 1885 Sophie Bryant First woman to have a paper published in LMS Proceedings 1916 Return to Burlington House the home of LMS until 1998 1937 ACE ’s Automatic Turing LMS Proceedings, 1937 Computing Engine, published Alan Turing’s first paper 1950 On Computable Numbers 1947 Death of G.H. -
TRINITY COLLEGE Cambridge Trinity College Cambridge College Trinity Annual Record Annual
2016 TRINITY COLLEGE cambridge trinity college cambridge annual record annual record 2016 Trinity College Cambridge Annual Record 2015–2016 Trinity College Cambridge CB2 1TQ Telephone: 01223 338400 e-mail: [email protected] website: www.trin.cam.ac.uk Contents 5 Editorial 11 Commemoration 12 Chapel Address 15 The Health of the College 18 The Master’s Response on Behalf of the College 25 Alumni Relations & Development 26 Alumni Relations and Associations 37 Dining Privileges 38 Annual Gatherings 39 Alumni Achievements CONTENTS 44 Donations to the College Library 47 College Activities 48 First & Third Trinity Boat Club 53 Field Clubs 71 Students’ Union and Societies 80 College Choir 83 Features 84 Hermes 86 Inside a Pirate’s Cookbook 93 “… Through a Glass Darkly…” 102 Robert Smith, John Harrison, and a College Clock 109 ‘We need to talk about Erskine’ 117 My time as advisor to the BBC’s War and Peace TRINITY ANNUAL RECORD 2016 | 3 123 Fellows, Staff, and Students 124 The Master and Fellows 139 Appointments and Distinctions 141 In Memoriam 155 A Ninetieth Birthday Speech 158 An Eightieth Birthday Speech 167 College Notes 181 The Register 182 In Memoriam 186 Addresses wanted CONTENTS TRINITY ANNUAL RECORD 2016 | 4 Editorial It is with some trepidation that I step into Boyd Hilton’s shoes and take on the editorship of this journal. He managed the transition to ‘glossy’ with flair and panache. As historian of the College and sometime holder of many of its working offices, he also brought a knowledge of its past and an understanding of its mysteries that I am unable to match. -
4. a Close Call: How a Near Failure Propelled Me to Succeed By
Early Career When I entered graduate study at Princeton, I brought A Close Call: How a my study habits (or lack thereof) with me. At the time in Princeton, the graduate classes did not have any home- Near Failure Propelled work or tests; the only major examination one had to pass (apart from some fairly easy language requirements) were Me to Succeed the dreaded “generals’’—the oral qualifying exams, often lasting over two hours, that one would take in front of Terence Tao three faculty members, usually in one’s second year. The questions would be drawn from five topics: real analysis, For as long as I can remember, I was always fascinated by complex analysis, algebra, and two topics of the student’s numbers and the formal symbolic operations of mathe- choice. For most of the other graduate students in my year, matics, even before I knew the uses of mathematics in the preparing for the generals was a top priority; they would real world. One of my earliest childhood memories was read textbooks from cover to cover, organise study groups, demanding that my grandmother, who was washing the and give each other mock exams. It had become a tradition windows, put detergent on the windows in the shape of for every graduate student taking the generals to write up numbers. When I was particularly rowdy as a child, my the questions they received and the answers they gave for parents would sometimes give me a math workbook to future students to practice. There were even skits performed work on instead, which I was more than happy to do. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
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 -
LMS Prize Winners
id745843 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com De Morgan House 57-58 Russell S quare London WC1B 4HS Media release 22 June 2007 Embargoed until 15:30 hrs Mathematician honoured for his $1 million dollar question The London Mathematical Society has awarded its foremost prize, the De Morgan Medal, to a mathematician who developed a problem which has yet to be solved. The LMS Council announced that Professor Bryan Birch, of the University of Oxford, had been awarded the triennial prize in recognition of his influential contributions to modern number theory. This area of mathematics is used in ensuring the security of Pin numbers and communications generally. In particular, Professor Birch worked with Professor Sir Peter Swinnerton-Dyer, of the University of Cambridge, to create a new area of arithmetic algebraic geometry. Together they formulated the Birch-Swinnerton-Dyer conjectures. Despite the best efforts of some of the greatest mathematical minds these remarkable conjectures are still open after 40 years and are amongst seven classic unsolved mathematical problems identified by the Clay Mathematics Institute in Cambridge, Massachusetts. The Institute is offering $1 million prizes for their proofs. Professor John Toland, President of the LMS, said, “These remarkable conjectures are still open and a million dollar prize awaits anyone who comes up with a proof. For these conjectures, and for many other seminal contributions, we honour Professor Birch today.” Professor Toland quoted mathematician Georg Cantor, who, in 1867 said, “In mathematics, the art of proposing a question must be held of higher value than solving it.” The Council for the LMS announced the De Morgan Medal and other prize winners at the Society Meeting held at University College London on 22 June. -
Professor Richard Courant (1888 – 1972)
Professor Richard Courant (1888 – 1972) From Wikipedia, the free encyclopedia, http://en.wikipedia.org/wiki/Richard_Courant Field: Mathematics Institutions: University of Göttingen University of Münster University of Cambridge New York University Alma mater: University of Göttingen Doctoral advisor: David Hilbert Doctoral students: William Feller Martin Kruskal Joseph Keller Kurt Friedrichs Hans Lewy Franz Rellich Known for: Courant number Courant minimax principle Courant–Friedrichs–Lewy condition Biography: Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there. As he found the courses not demanding enough, he continued his studies in Zürich and Göttingen. Courant eventually became David Hilbert's assistant in Göttingen and obtained his doctorate there in 1910. He had to fight in World War I, but he was wounded and dismissed from the military service shortly after enlisting. After the war, in 1919, he married Nerina (Nina) Runge, a daughter of the Göttingen professor for Applied Mathematics, Carl Runge. Richard continued his research in Göttingen, with a two-year period as professor in Münster. There he founded the Mathematical Institute, which he headed as director from 1928 until 1933. Courant left Germany in 1933, earlier than many of his colleagues. While he was classified as a Jew by the Nazis, his having served as a front-line soldier exempted him from losing his position for this particular reason at the time; however, his public membership in the social-democratic left was a reason for dismissal to which no such exemption applied.[1] After one year in Cambridge, Courant went to New York City where he became a professor at New York University in 1936. -
Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- Mogenization
in \Homogenization 2001, Proceedings of the First HMS2000 International School and Conference on Ho- mogenization. Naples, Complesso Monte S. Angelo, June 18-22 and 23-27, 2001, Ed. L. Carbone and R. De Arcangelis, 191{211, Gakkotosho, Tokyo, 2003". On Homogenization and Γ-convergence Luc TARTAR1 In memory of Ennio DE GIORGI When in the Fall of 1976 I had chosen \Homog´en´eisationdans les ´equationsaux d´eriv´eespartielles" (Homogenization in partial differential equations) for the title of my Peccot lectures, which I gave in the beginning of 1977 at Coll`egede France in Paris, I did not know of the term Γ-convergence, which I first heard almost a year after, in a talk that Ennio DE GIORGI gave in the seminar that Jacques-Louis LIONS was organizing at Coll`egede France on Friday afternoons. I had not found the definition of Γ-convergence really new, as it was quite similar to questions that were already discussed in control theory under the name of relaxation (which was a more general question than what most people mean by that term now), and it was the convergence in the sense of Umberto MOSCO [Mos] but without the restriction to convex functionals, and it was the natural nonlinear analog of a result concerning G-convergence that Ennio DE GIORGI had obtained with Sergio SPAGNOLO [DG&Spa]; however, Ennio DE GIORGI's talk contained a quite interesting example, for which he referred to Luciano MODICA (and Stefano MORTOLA) [Mod&Mor], where functionals involving surface integrals appeared as Γ-limits of functionals involving volume integrals, and I thought that it was the interesting part of the concept, so I had found it similar to previous questions but I had felt that the point of view was slightly different. -
Linking Together Members of the Mathematical Carlos Rocha, University of Lisbon; Jean Taylor, Cour- Community from the US and Abroad
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Features Epimorphism Theorem Prime Numbers Interview J.-P. Bourguignon Societies European Physical Society Research Centres ESI Vienna December 2013 Issue 90 ISSN 1027-488X S E European M M Mathematical E S Society Cover photo: Jean-François Dars Mathematics and Computer Science from EDP Sciences www.esaim-cocv.org www.mmnp-journal.org www.rairo-ro.org www.esaim-m2an.org www.esaim-ps.org www.rairo-ita.org Contents Editorial Team European Editor-in-Chief Ulf Persson Matematiska Vetenskaper Lucia Di Vizio Chalmers tekniska högskola Université de Versailles- S-412 96 Göteborg, Sweden St Quentin e-mail: [email protected] Mathematical Laboratoire de Mathématiques 45 avenue des États-Unis Zdzisław Pogoda 78035 Versailles cedex, France Institute of Mathematicsr e-mail: [email protected] Jagiellonian University Society ul. prof. Stanisława Copy Editor Łojasiewicza 30-348 Kraków, Poland Chris Nunn e-mail: [email protected] Newsletter No. 90, December 2013 119 St Michaels Road, Aldershot, GU12 4JW, UK Themistocles M. Rassias Editorial: Meetings of Presidents – S. Huggett ............................ 3 e-mail: [email protected] (Problem Corner) Department of Mathematics A New Cover for the Newsletter – The Editorial Board ................. 5 Editors National Technical University Jean-Pierre Bourguignon: New President of the ERC .................. 8 of Athens, Zografou Campus Mariolina Bartolini Bussi GR-15780 Athens, Greece Peter Scholze to Receive 2013 Sastra Ramanujan Prize – K. Alladi 9 (Math. Education) e-mail: [email protected] DESU – Universitá di Modena e European Level Organisations for Women Mathematicians – Reggio Emilia Volker R. Remmert C. Series ............................................................................... 11 Via Allegri, 9 (History of Mathematics) Forty Years of the Epimorphism Theorem – I-42121 Reggio Emilia, Italy IZWT, Wuppertal University [email protected] D-42119 Wuppertal, Germany P.