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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. -
CURRENT EVENTS BULLETIN Friday, January 8, 2016, 1:00 PM to 5:00 PM Room 4C-3 Washington State Convention Center Joint Mathematics Meetings, Seattle, WA
A MERICAN M ATHEMATICAL S OCIETY CURRENT EVENTS BULLETIN Friday, January 8, 2016, 1:00 PM to 5:00 PM Room 4C-3 Washington State Convention Center Joint Mathematics Meetings, Seattle, WA 1:00 PM Carina Curto, Pennsylvania State University What can topology tell us about the neural code? Surprising new applications of what used to be thought of as “pure” mathematics. 2:00 PM Yuval Peres, Microsoft Research and University of California, Berkeley, and Lionel Levine, Cornell University Laplacian growth, sandpiles and scaling limits Striking large-scale structure arising from simple cellular automata. 3:00 PM Timothy Gowers, Cambridge University Probabilistic combinatorics and the recent work of Peter Keevash The major existence conjecture for combinatorial designs has been proven! 4:00 PM Amie Wilkinson, University of Chicago What are Lyapunov exponents, and why are they interesting? A basic tool in understanding the predictability of physical systems, explained. Organized by David Eisenbud, Mathematical Sciences Research Institute Introduction to the Current Events Bulletin Will the Riemann Hypothesis be proved this week? What is the Geometric Langlands Conjecture about? How could you best exploit a stream of data flowing by too fast to capture? I think we mathematicians are provoked to ask such questions by our sense that underneath the vastness of mathematics is a fundamental unity allowing us to look into many different corners -- though we couldn't possibly work in all of them. I love the idea of having an expert explain such things to me in a brief, accessible way. And I, like most of us, love common-room gossip. -
Abstract We Survey the Published Work of Harry Kesten in Probability Theory, with Emphasis on His Contributions to Random Walks
Abstract We survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, perco- lation, and related topics. Keywords Probability, random walk, branching process, random matrix, diffusion limited aggregation, percolation. Mathematics Subject Classification (2010) 60-03, 60G50, 60J80, 60B20, 60K35, 82B20. Noname manuscript No. 2(will be inserted by the editor) Geoffrey R. Grimmett Harry Kesten's work in probability theory Geoffrey R. Grimmett In memory of Harry Kesten, inspiring colleague, valued friend April 8, 2020 1 Overview Harry Kesten was a prominent mathematician and personality in a golden period of probability theory from 1956 to 2018. At the time of Harry's move from the Netherlands to the USA in 1956, as a graduate student aged 24, much of the foundational infrastructure of probability was in place. The central characters of probability had long been identified (including random walk, Brownian motion, the branching process, and the Poisson process), and connections had been made and developed between `pure theory' and cognate areas ranging from physics to finance. In the half-century or so since 1956, a coordinated and refined theory has been developed, and probability has been recognised as a crossroads discipline in mathematical science. Few mathematicians have contributed as much during this period as Harry Kesten. Following a turbulent childhood (see [59]), Harry studied mathematics with David van Dantzig and Jan Hemelrijk in Amsterdam, where in 1955 he attended a lecture by Mark Kac entitled \Some probabilistic aspects of potential theory". This encounter appears to have had a decisive effect, in that Harry moved in 1956 to Cornell University to work with Kac. -
Interviewed by T. Christine Stevens)
KENNETH A. ROSS JANUARY 8, 2011 AND JANUARY 5, 2012 (Interviewed by T. Christine Stevens) How did you get involved in the MAA? As a good citizen of the mathematical community, I was a member of MAA from the beginning of my career. But I worked in an “AMS culture,” so I wasn’t actively involved in the MAA. As of January, 1983, I had never served on an MAA committee. But I had been Associate Secretary of the AMS from 1971 to 1981, and thus Len Gillman (who was MAA Treasurer at the time) asked me to be MAA Secretary. There was a strong contrast between the cultures of the AMS and the MAA, and my first two years were very hard. Did you receive mentoring in the MAA at the early stages of your career? From whom? As a graduate student at the University of Washington, I hadn’t even been aware that the department chairman, Carl Allendoerfer, was serving at the time as MAA President. My first mentor in the MAA was Len Gillman, who got me involved with the MAA. Being Secretary and Treasurer, respectively, we consulted a lot, and he was the one who helped me learn the MAA culture. One confession: At that time, approvals for new unbudgeted expenses under $500 were handled by the Secretary, the Treasurer and the Executive Director, Al Wilcox. The requests usually came to me first. Since Len was consistently tough, and Al was a push-over, I would first ask the one whose answer would agree with mine, and then with a 2-0 vote, I didn’t have to even bother the other one. -
What Are Lyapunov Exponents, and Why Are They Interesting?
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 1, January 2017, Pages 79–105 http://dx.doi.org/10.1090/bull/1552 Article electronically published on September 6, 2016 WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING? AMIE WILKINSON Introduction At the 2014 International Congress of Mathematicians in Seoul, South Korea, Franco-Brazilian mathematician Artur Avila was awarded the Fields Medal for “his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.”1 Although it is not explicitly mentioned in this citation, there is a second unify- ing concept in Avila’s work that is closely tied with renormalization: Lyapunov (or characteristic) exponents. Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odinger operators. Here we take the op- portunity to explore these areas and reveal some underlying themes connecting exponents, chaotic dynamics and renormalization. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats: the iterated barycentric subdivision of a triangle. When the midpoint of each side of a triangle is connected to its opposite vertex by a line segment, the three resulting segments meet in a point in the interior of the triangle. The barycentric subdivision of a triangle is the collection of 6 smaller triangles determined by these segments and the edges of the original triangle: Figure 1. Barycentric subdivision. Received by the editors August 2, 2016. -
Sir Andrew J. Wiles
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society March 2017 Volume 64, Number 3 Women's History Month Ad Honorem Sir Andrew J. Wiles page 197 2018 Leroy P. Steele Prize: Call for Nominations page 195 Interview with New AMS President Kenneth A. Ribet page 229 New York Meeting page 291 Sir Andrew J. Wiles, 2016 Abel Laureate. “The definition of a good mathematical problem is the mathematics it generates rather Notices than the problem itself.” of the American Mathematical Society March 2017 FEATURES 197 239229 26239 Ad Honorem Sir Andrew J. Interview with New The Graduate Student Wiles AMS President Kenneth Section Interview with Abel Laureate Sir A. Ribet Interview with Ryan Haskett Andrew J. Wiles by Martin Raussen and by Alexander Diaz-Lopez Allyn Jackson Christian Skau WHAT IS...an Elliptic Curve? Andrew Wiles's Marvelous Proof by by Harris B. Daniels and Álvaro Henri Darmon Lozano-Robledo The Mathematical Works of Andrew Wiles by Christopher Skinner In this issue we honor Sir Andrew J. Wiles, prover of Fermat's Last Theorem, recipient of the 2016 Abel Prize, and star of the NOVA video The Proof. We've got the official interview, reprinted from the newsletter of our friends in the European Mathematical Society; "Andrew Wiles's Marvelous Proof" by Henri Darmon; and a collection of articles on "The Mathematical Works of Andrew Wiles" assembled by guest editor Christopher Skinner. We welcome the new AMS president, Ken Ribet (another star of The Proof). Marcelo Viana, Director of IMPA in Rio, describes "Math in Brazil" on the eve of the upcoming IMO and ICM. -
Mori Dream Spaces and Birational Rigidity of Fano 3-Folds
Ahmadinezhad, H., & Zucconi, F. (2016). Mori dream spaces and birational rigidity of Fano 3-folds. Advances in Mathematics, 292, 410- 445. https://doi.org/10.1016/j.aim.2016.01.008 Peer reviewed version Link to published version (if available): 10.1016/j.aim.2016.01.008 Link to publication record in Explore Bristol Research PDF-document University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ MORI DREAM SPACES AND BIRATIONAL RIGIDITY OF FANO 3-FOLDS HAMID AHMADINEZHAD AND FRANCESCO ZUCCONI Abstract. We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces. 1. Introduction Rationality question for varieties covered by rational curves (e.g. unirational varieties) has been a fundamental problem in algebraic geometry, the answer to which has opened various subjects across the field. The first examples of unirational varieties that are irrational were found about the same time by Clemens and Griffiths [10], Iskovskikh and Manin [18], Artin and Mumford [3]. While the results of [10] and [3] indicate that the varieties under study (respectively, a smooth cubic 3-fold and some special singular quartic 3-folds in the weighted projective space P(1; 1; 1; 1; 2)) are irrational, the result in [18] shows that a smooth quartic 3-fold is not birational to any conic bundle or fibration into del 3 Pezzo surfaces, or any other Fano variety, including P , that is to say it is irrational in a strong sense. -
Richard Schoen – Mathematics
Rolf Schock Prizes 2017 Photo: Private Photo: Richard Schoen Richard Schoen – Mathematics The Rolf Schock Prize in Mathematics 2017 is awarded to Richard Schoen, University of California, Irvine and Stanford University, USA, “for groundbreaking work in differential geometry and geometric analysis including the proof of the Yamabe conjecture, the positive mass conjecture, and the differentiable sphere theorem”. Richard Schoen holds professorships at University of California, Irvine and Stanford University, and is one of three vice-presidents of the American Mathematical Society. Schoen works in the field of geometric analysis. He is in fact together with Shing-Tung Yau one of the founders of the subject. Geometric analysis can be described as the study of geometry using non-linear partial differential equations. The developments in and around this field has transformed large parts of mathematics in striking ways. Examples include, gauge theory in 4-manifold topology, Floer homology and Gromov-Witten theory, and Ricci-and mean curvature flows. From the very beginning Schoen has produced very strong results in the area. His work is characterized by powerful technical strength and a clear vision of geometric relevance, as demonstrated by him being involved in the early stages of areas that later witnessed breakthroughs. Examples are his work with Uhlenbeck related to gauge theory and his work with Simon and Yau, and with Yau on estimates for minimal surfaces. Schoen has also established a number of well-known and classical results including the following: • The positive mass conjecture in general relativity: the ADM mass, which measures the deviation of the metric tensor from the imposed flat metric at infinity is non-negative. -
2008 Award for Distinguished Public Service
2008 Award for Distinguished Public Service The 2008 Award for Distinguished and continuation of the Park City/IAS Mathemat- Public Service was presented at the ics Institute. 114th Annual Meeting of the AMS in San Diego in January 2008. Biographical Sketch The Award for Distinguished Public Herbert Clemens earned his Ph.D. in 1966 from Service is presented every two years the University of California, Berkeley, under the to a research mathematician who has direction of Phillip A. Griffiths. He has taught at made a distinguished contribution to Columbia University, the University of Utah, and the mathematics profession during the preceding five years. The purpose of the Ohio State University, where he has been on the award is to encourage and recog- the faculty since 2002. He has served as director nize those individuals who contribute of the NSF Regional Geometry Institute, Park City, their time to public service activities UT, and chair of the Steering Committee for the IAS in support of mathematics. The award Park City Mathematics Institute. He was an invited Herbert Clemens carries a cash prize of US$4,000. speaker at the International Congress of Mathema- The Award for Distinguished Public ticians in 1974 and in 1986. His academic honors Service is made by the AMS Council include a Silver Medal from the Italian Mathemati- acting on the recommendation of a selection com- cal Society and a Laurea de honoris causa from the mittee. For the 2008 award the members of the Universita di Torino, among others. His research selection committee were: William J. Lewis, Carolyn R. -
Clay Mathematics Institute 2005 James A
Contents Clay Mathematics Institute 2005 James A. Carlson Letter from the President 2 The Prize Problems The Millennium Prize Problems 3 Recognizing Achievement 2005 Clay Research Awards 4 CMI Researchers Summary of 2005 6 Workshops & Conferences CMI Programs & Research Activities Student Programs Collected Works James Arthur Archive 9 Raoul Bott Library CMI Profile Interview with Research Fellow 10 Maria Chudnovsky Feature Article Can Biology Lead to New Theorems? 13 by Bernd Sturmfels CMI Summer Schools Summary 14 Ricci Flow, 3–Manifolds, and Geometry 15 at MSRI Program Overview CMI Senior Scholars Program 17 Institute News Euclid and His Heritage Meeting 18 Appointments & Honors 20 CMI Publications Selected Articles by Research Fellows 27 Books & Videos 28 About CMI Profile of Bow Street Staff 30 CMI Activities 2006 Institute Calendar 32 2005 Euclid: www.claymath.org/euclid James Arthur Collected Works: www.claymath.org/cw/arthur Hanoi Institute of Mathematics: www.math.ac.vn Ramanujan Society: www.ramanujanmathsociety.org $.* $MBZ.BUIFNBUJDT*OTUJUVUF ".4 "NFSJDBO.BUIFNBUJDBM4PDJFUZ In addition to major,0O"VHVTU BUUIFTFDPOE*OUFSOBUJPOBM$POHSFTTPG.BUIFNBUJDJBOT ongoing activities such as JO1BSJT %BWJE)JMCFSUEFMJWFSFEIJTGBNPVTMFDUVSFJOXIJDIIFEFTDSJCFE the summer schools,UXFOUZUISFFQSPCMFNTUIBUXFSFUPQMBZBOJOnVFOUJBMSPMFJONBUIFNBUJDBM the Institute undertakes a 5IF.JMMFOOJVN1SJ[F1SPCMFNT SFTFBSDI"DFOUVSZMBUFS PO.BZ BUBNFFUJOHBUUIF$PMMÒHFEF number of smaller'SBODF UIF$MBZ.BUIFNBUJDT*OTUJUVUF $.* BOOPVODFEUIFDSFBUJPOPGB special projects -
Committee to Select the Winner of the Veblen Prize
Committee to Select the Winner of the Veblen Prize General Description · Committee is standing · Number of members is three · A new committee is appointed for each award Principal Activities The award is made every three years at the Annual (January) Meeting, in a cycle including the years 2001, 2004, and so forth. The award is made for a notable research work in geometry or topology that has appeared in the last six years. The work must be published in a recognized, peer-reviewed venue. The committee recommends a winner and communicates this recommendation to the Secretary for approval by the Executive Committee of the Council. Past winners are listed in the November issue of the Notices in odd-numbered years. About this Prize This prize was established in 1961 in memory of Professor Oswald Veblen through a fund contributed by former students and colleagues. The fund was later doubled by a gift from Elizabeth Veblen, widow of Professor Veblen. In honor of her late father, John L. Synge, who knew and admired Oswald Veblen, Cathleen Synge Morawetz and her husband, Herbert, substantially increased the endowed fund in 2013. The award is supplemented by a Steele Prize from the AMS under the name of the Veblen Prize. The current prize amount is $5,000. Miscellaneous Information The business of this committee can be done by mail, electronic mail, or telephone, expenses which may be reimbursed by the Society. Note to the Chair Work done by committees on recurring problems may have value as precedent or work done may have historical interest. -
Curriculum Vitae.Pdf
Lan-Hsuan Huang Department of Mathematics Phone: (860) 486-8390 University of Connecticut Fax: (860) 486-4238 Storrs, CT 06269 Email: [email protected] USA http://lhhuang.math.uconn.edu Research Geometric Analysis and General Relativity Employment University of Connecticut Professor 2020-present Associate Professor 2016-2020 Assistant Professor 2012-2016 Institute for Advanced Study Member (with the title of von Neumann fellow) 2018-2019 Columbia University Ritt Assistant Professor 2009-2012 Education Ph.D. Mathematics, Stanford University 2009 Advisor: Professor Richard Schoen B.S. Mathematics, National Taiwan University 2004 Grants • NSF DMS-2005588 (PI, $250,336) 2020-2023 & Honors • von Neumann Fellow, Institute for Advanced Study 2018-2019 • Simons Fellow in Mathematics, Simons Foundation ($122,378) 2018-2019 • NSF CAREER Award (PI, $400,648) 2015-2021 • NSF Grant DMS-1308837 (PI, $282,249) 2013-2016 • NSF Grant DMS-1005560 and DMS-1301645 (PI, $125,645) 2010-2013 Visiting • Erwin Schr¨odingerInternational Institute July 2017 Positions • National Taiwan University Summer 2016 • MSRI Research Member Fall 2013 • Max-Planck Institute for Gravitational Physics, Germany Fall 2010 • Institut Mittag-Leffler, Sweden Fall 2008 1 Journal 1. Equality in the spacetime positive mass theorem (with D. Lee), Commu- Publications nications in Mathematical Physics 376 (2020), no. 3, 2379{2407. 2. Mass rigidity for hyperbolic manifolds (with H. C. Jang and D. Martin), Communications in Mathematical Physics 376 (2020), no. 3, 2329- 2349. 3. Localized deformation for initial data sets with the dominant energy condi- tion (with J. Corvino), Calculus Variations and Partial Differential Equations (2020), no. 1, No. 42. 4. Existence of harmonic maps into CAT(1) spaces (with C.