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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. -
Caucher Birkar — from Asylum Seeker to Fields Medal Winner at Cambridge
MATHS, 1 Caucher Birkar, 41, at VERSION Cambridge University, photographed by Jude Edginton REPR O OP HEARD THE ONE ABOUT THE ASYLUM SEEKER SUBS WHO WANDERED INTO A BRITISH UNIVERSITY... A RT AND CAME OUT A MATHS SUPERSTAR? PR ODUCTION CLIENT Caucher Birkar grew up in a Kurdish peasant family in a war zone and arrived in Nottingham as a refugee – now he has received the mathematics equivalent of the Nobel prize. By Tom Whipple BLACK YELLOW MAGENTA CYAN 91TTM1940232.pgs 01.04.2019 17:39 MATHS, 2 VERSION ineteen years ago, the mathematics Caucher Birkar in Isfahan, Receiving the Fields Medal If that makes sense, congratulations: you department at the University of Iran, in 1999 in Rio de Janeiro, 2018 now have a very hazy understanding of Nottingham received an email algebraic geometry. This is the field that from an asylum seeker who Birkar works in. wanted to talk to someone about The problem with explaining maths is REPR algebraic geometry. not, or at least not always, the stupidity of his They replied and invited him in. listeners. It is more fundamental than that: O OP N So it was that, shortly afterwards, it is language. Mathematics is not designed Caucher Birkar, the 21-year-old to be described in words. It is designed to be son of a Kurdish peasant family, described in mathematics. This is the great stood in front of Ivan Fesenko, a professor at triumph of the subject. It was why a Kurdish Nottingham, and began speaking in broken asylum seeker with bad English could convince SUBS English. -
Birational Geometry of Algebraic Varieties
Proc. Int. Cong. of Math. – 2018 Rio de Janeiro, Vol. 1 (563–588) BIRATIONAL GEOMETRY OF ALGEBRAIC VARIETIES Caucher Birkar 1 Introduction This is a report on some of the main developments in birational geometry in recent years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero. This is not a comprehensive survey of all advances in birational geometry, e.g. we will not touch upon the positive characteristic case which is a very active area of research. We will work over an algebraically closed field k of characteristic zero. Varieties are all quasi-projective. Birational geometry, with the so-called minimal model program at its core, aims to classify algebraic varieties up to birational isomorphism by identifying “nice” elements in each birational class and then classifying such elements, e.g study their moduli spaces. Two varieties are birational if they contain isomorphic open subsets. In dimension one, a nice element in a birational class is simply a smooth and projective element. In higher dimension though there are infinitely many such elements in each class, so picking a rep- resentative is a very challenging problem. Before going any further lets introduce the canonical divisor. 1.1 Canonical divisor. To understand a variety X one studies subvarieties and sheaves on it. Subvarieties of codimension one and their linear combinations, that is, divisors play a crucial role. Of particular importance is the canonical divisor KX . When X is smooth this is the divisor (class) whose associated sheaf OX (KX ) is the canonical sheaf !X := det ΩX where ΩX is the sheaf of regular differential forms. -
Eigenvalue Distributions of Beta-Wishart Matrices
Eigenvalue distributions of beta-Wishart matrices The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Edelman, Alan, and Plamen Koev. “Eigenvalue Distributions of Beta- Wishart Matrices.” Random Matrices: Theory and Applications 03, no. 02 (April 2014): 1450009. As Published http://dx.doi.org/10.1142/S2010326314500099 Publisher World Scientific Pub Co Pte Lt Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/116006 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ SIAM J. MATRIX ANAL. APPL. c 2013 Society for Industrial and Applied Mathematics Vol. XX, No. X, pp. XX{XX EIGENVALUE DISTRIBUTIONS OF BETA-WISHART MATRICES∗ ALAN EDELMANy AND PLAMEN KOEVz Abstract. We derive explicit expressions for the distributions of the extreme eigenvalues of the Beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results generalize the classical results for the real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart matrices to any β > 0. Key words. random matrix, Wishart distribution, eigenavalue, hypergeometric function of a matrix argument AMS subject classifications. 15A52, 60E05, 62H10, 65F15 DOI. XX.XXXX/SXXXXXXXXXXXXXXXX 1. Introduction. Recently, the classical real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart random matrix ensembles were generalized to any β > 0 by what is now called the Beta{Wishart ensemble [2, 9]. In this paper we derive the explicit distributions for the extreme eigenvalues and the trace of this ensemble as series of Jack functions and, in particular, in terms of the hypergeometric function of matrix argument. -
Pterossauros Nasciam Sem a Capacidade De Voar
NOTAS 1 Pterossauros nasciam sem a capacidade de voar Os filhotes de pterossauros, répteis ala- dos já extintos, contemporâneos dos dinossauros, rompiam seus ovos prontos para andar, mas não para bater asas e ga- nhar os ares. Assim que nasciam, os ossos Ovos fossilizados da cintura estavam formados. Isso permi- do pterossauro Hamipterus tia que eles se apoiassem sobre as patas tianshanensis e traseiras e dessem os primeiros passos. reconstituição Porém, a estrutura óssea que dá suporte artística da espécie aos movimentos do músculo peitoral, essencial para sustentar o voo, ainda não estava totalmente constituída. Os recém- -nascidos também não tinham todos os dentes, limitação que provavelmente os impedia de se alimentar sozinhos. Para sobreviver até que os ossos de apoio das asas e os dentes estivessem completos, os filhotes tinham de permanecer um bom tempo sob o cuidado dos pais. Esse cenário, sobre o desenvolvimento embrio- nário e os primeiros movimentos, ainda tímidos, dos filhotes de pterossauros, é sugerido em um estudo feito por paleon- tólogos brasileiros e chineses (Science, 1º de dezembro). “O descompasso entre o desenvolvimento dos ossos da cintura e os da musculatura peitoral indica que os pterossauros não conseguiam voar ao nascer”, comenta o paleontólogo Ale- xander Kellner, do Museu Nacional da Universidade Federal do Rio de Janeiro (UFRJ), um dos autores do trabalho. Com o auxílio de imagens de tomografia com- putadorizada, o grupo analisou o interior de 16 ovos de pterossauros da espécie Hamipterus tianshanensis, que viveu há cerca de 120 milhões de anos na bacia de Turpan-Hami, no noroeste da China. -
Matrix Algorithms: Fast, Stable, Communication-Optimizing—Random?!
Matrix algorithms: fast, stable, communication-optimizing—random?! Ioana Dumitriu Department of Mathematics University of Washington (Seattle) Joint work with Grey Ballard, James Demmel, Olga Holtz, Robert Kleinberg IPAM: Convex Optimization and Algebraic Geometry September 28, 2010 Ioana Dumitriu (UW) Matrix algorithms September 28, 2010 1 / 49 1 Motivation and Background Main Concerns in Numerical Linear Algebra Size issues 2 Reducing the flops Stability of FMM Stability of FLA Why Randomize? 3 Reducing communication Why is communication bad? Randomized Spectral Divide and Conquer Communication Costs 4 Conclusions Ioana Dumitriu (UW) Matrix algorithms September 28, 2010 2 / 49 Motivation and Background Main Concerns in Numerical Linear Algebra The big four issues in matrix algorithms. Accuracy of computation (how “far” the computed values are from the actual ones, in the presence of rounding error). Stability (how much the computed result will change if we perturb the problem a little bit). Speed/complexity (how many operations they require, e.g., multiplications). Parallelism (how to distribute the computation to multiple processors in order to optimize speed – if possible). Communication complexity (how to minimize the amount of back-and-forth between levels of memory or processors.) Ioana Dumitriu (UW) Matrix algorithms September 28, 2010 3 / 49 Motivation and Background Main Concerns in Numerical Linear Algebra The big four issues in matrix algorithms. Accuracy of computation (how “far” the computed values are from the actual ones, in the presence of rounding error). Stability (how much the computed result will change if we perturb the problem a little bit). Speed/flops complexity (how many operations they require, e.g., multiplications). -
Looking at Earth: an Astronaut's Journey Induction Ceremony 2017
american academy of arts & sciences winter 2018 www.amacad.org Bulletin vol. lxxi, no. 2 Induction Ceremony 2017 Class Speakers: Jane Mayer, Ursula Burns, James P. Allison, Heather K. Gerken, and Gerald Chan Annual David M. Rubenstein Lecture Looking at Earth: An Astronaut’s Journey David M. Rubenstein and Kathryn D. Sullivan ALSO: How Are Humans Different from Other Great Apes?–Ajit Varki, Pascal Gagneux, and Fred H. Gage Advancing Higher Education in America–Monica Lozano, Robert J. Birgeneau, Bob Jacobsen, and Michael S. McPherson Redistricting and Representation–Patti B. Saris, Gary King, Jamal Greene, and Moon Duchin noteworthy Select Prizes and Andrea Bertozzi (University of James R. Downing (St. Jude Chil- Barbara Grosz (Harvard Univer- California, Los Angeles) was se- dren’s Research Hospital) was sity) is the recipient of the Life- Awards to Members lected as a 2017 Simons Investi- awarded the 2017 E. Donnall time Achievement Award of the gator by the Simons Foundation. Thomas Lecture and Prize by the Association for Computational American Society of Hematology. Linguistics. Nobel Prize in Chemistry, Clara D. Bloomfield (Ohio State 2017 University) is the recipient of the Carol Dweck (Stanford Univer- Christopher Hacon (University 2017 Robert A. Kyle Award for sity) was awarded the inaugural of Utah) was awarded the Break- Joachim Frank (Columbia Univer- Outstanding Clinician-Scientist, Yidan Prize. through Prize in Mathematics. sity) presented by the Mayo Clinic Di- vision of Hematology. Felton Earls (Harvard Univer- Naomi Halas (Rice University) sity) is the recipient of the 2018 was awarded the 2018 Julius Ed- Nobel Prize in Economic Emmanuel J. -
The Random Matrix Technique of Ghosts and Shadows
The Random Matrix Technique of Ghosts and Shadows Alan Edelman November 22, 2009 Abstract We propose to abandon the notion that a random matrix has to be sampled for it to exist. Much of today's applied nite random matrix theory concerns real or complex random matrices (β = 1; 2). The threefold way so named by Dyson in 1962 [2] adds quaternions (β = 4). While it is true there are only three real division algebras (β=dimension over the reals), this mathematical fact while critical in some ways, in other ways is irrelevant and perhaps has been over interpreted over the decades. We introduce the notion of a ghost random matrix quantity that exists for every beta, and a shadow quantity which may be real or complex which allows for computation. Any number of computations have successfully given reasonable answers to date though diculties remain in some cases. Though it may seem absurd to have a three and a quarter dimensional or pi dimensional algebra, that is exactly what we propose and what we compute with. In the end β becomes a noisiness parameter rather than a dimension. 1 Introduction This conference article contains an idea which has become a technique. Perhaps it might be labeled a conjecture, but I think idea is the better label right now. The idea was discussed informally to a number of researchers and students at MIT for a number of years now, probably dating back to 2003 or so. It was also presented at a number of conferences [3] . As slides do not quite capture a talk, this seemed a good place to write down the ideas. -
Mathematics Calendar
Mathematics Calendar Please submit conference information for the Mathematics Calendar through the Mathematics Calendar submission form at http:// www.ams.org/cgi-bin/mathcal-submit.pl. The most comprehensive and up-to-date Mathematics Calendar information is available on the AMS website at http://www.ams.org/mathcal/. June 2014 Information: http://www.tesol.org/attend-and-learn/ online-courses-seminars/esl-for-the-secondary- 1–7 Modern Time-Frequency Analysis, Strobl, Austria. (Apr. 2013, mathematics-teacher. p. 429) * 2–30 Algorithmic Randomness, Institute for Mathematical Sciences, 2–5 WSCG 2014 - 22nd International Conference on Computer National University of Singapore, Singapore. Graphics, Visualization and Computer Vision 2014, Primavera Description: Activities 1. Informal Collaboration: June 2–8, 2014. 2. Hotel and Congress Centrum, Plzen (close to Prague), Czech Repub- Ninth International Conference on Computability, Complexity and lic. (Jan. 2014, p. 90) Randomness (CCR 2014): June 9–13, 2014. The conference series 2–6 AIM Workshop: Descriptive inner model theory, American “Computability, Complexity and Randomness” is centered on devel- Institute of Mathematics, Palo Alto, California. (Mar. 2014, p. 312) opments in Algorithmic Randomness, and the conference CCR 2014 2–6 Computational Nonlinear Algebra, Institute for Computational will be part of the IMS programme. The CCR has previously been held and Experimental Research in Mathematics, (ICERM), Brown Univer- in Cordoba 2004, in Buenos Aires 2007, in Nanjing 2008, in Luminy sity, Providence, Rhode Island. (Nov. 2013, p. 1398) 2009, in Notre Dame 2010, in Cape Town 2011, in Cambridge 2012, and in Moscow 2013; it will be held in Heidelberg 2015. 3. Informal 2–6 Conference on Ulam’s type stability, Rytro, Poland. -
2010 Integral
Autumn 2010 Volume 5 Massachusetts Institute of Technology 1ntegral n e w s f r o m t h e mathematics d e p a r t m e n t a t m i t The retirement of seven of our illustrious col- leagues this year—Mike Artin, David Ben- Inside ney, Dan Kleitman, Arthur Mattuck, Is Sing- er, Dan Stroock, and Alar Toomre—marks • Faculty news 2–3 a shift to a new generation of faculty, from • Women in math 3 those who entered the field in the Sputnik era to those who never knew life without email • A minority perspective 4 and the Internet. The older generation built • Student news 4–5 the department into the academic power- house it is today—indeed they were the core • Funds for RSI and SPUR 5 of the department, its leadership and most • Retiring faculty and staff 6–7 distinguished members, during my early years at MIT. Now, as they are in the process • Alumni corner 8 of retiring, I look around and see that my con- temporaries are becoming the department’s Dear Friends, older group. Yikes! another year gone by and what a year it Other big changes are in the works. Two of Marina Chen have taken the lead in raising an was. We’re getting older, and younger, cele- our dedicated long-term administrators— endowment for them. Together with those of brating prizes and long careers, remembering Joanne Jonsson and Linda Okun—have Tim Lu ’79 and Peiti Tung ’79, their commit- our past and looking to the future. -
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. -
Orbital Varieties and Unipotent Representations of Classical
Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups by Thomas Pietraho M.S., University of Chicago, 1996 B.A., University of Chicago, 1996 Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2001 °c Thomas Pietraho, MMI. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part and to grant others the right to do so. Author ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Department of Mathematics April 25, 2001 Certified by :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: David A. Vogan Professor of Mathematics Thesis Supervisor Accepted by :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Tomasz Mrowka Chairman, Department Committee on Graduate Students 2 Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups by Thomas Pietraho Submitted to the Department of Mathematics on April 25, 2001, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Let G be a complex semi-simple and classical Lie group. The notion of a Lagrangian covering can be used to extend the method of polarizing a nilpotent coadjoint orbit to obtain a unitary representation of G. W. Graham and D. Vogan propose such a construction, relying on the notions of orbital varieties and admissible orbit data. The first part of the thesis seeks to understand the set of orbital varieties contained in a given nipotent orbit. Starting from N. Spaltenstein’s parameterization of the irreducible components of the variety of flags fixed by a unipotent, we produce a parameterization of the orbital varieties lying in the corresponding fiber of the Steinberg map.