Interview of Top Finishers on the 2004 Putnam Exam
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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. -
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. -
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). -
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 of Data Science
SIAM JOURNAL ON Mathematics of Data Science Volume 2 • 2020 Editor-in-Chief Tamara G. Kolda, Sandia National Laboratories Section Editors Mark Girolami, University of Cambridge, UK Alfred Hero, University of Michigan, USA Robert D. Nowak, University of Wisconsin, Madison, USA Joel A. Tropp, California Institute of Technology, USA Associate Editors Maria-Florina Balcan, Carnegie Mellon University, USA Vianney Perchet, ENSAE, CRITEO, France Rina Foygel Barber, University of Chicago, USA Jonas Peters, University of Copenhagen, Denmark Mikhail Belkin, University of California, San Diego, USA Natesh Pillai, Harvard University, USA Robert Calderbank, Duke University, USA Ali Pinar, Sandia National Laboratories, USA Coralia Cartis, University of Oxford, UK Mason Porter, University of Califrornia, Los Angeles, USA Venkat Chandrasekaran, California Institute of Technology, Maxim Raginsky, University of Illinois, USA Urbana-Champaign, USA Patrick L. Combettes, North Carolina State University, USA Bala Rajaratnam, University of California, Davis, USA Alexandre d’Aspremont, CRNS, Ecole Normale Superieure, Philippe Rigollet, MIT, USA France Justin Romberg, Georgia Tech, USA Ioana Dumitriu, University of California, San Diego, USA C. Seshadhri, University of California, Santa Cruz, USA Maryam Fazel, University of Washington, USA Amit Singer, Princeton University, USA David F. Gleich, Purdue University, USA Marc Teboulle, Tel Aviv University, Israel Wouter Koolen, CWI, the Netherlands Caroline Uhler, MIT, USA Gitta Kutyniok, University of Munich, Germany -
2017-2018 Annual Report
Institute for Computational and Experimental Research in Mathematics Annual Report May 1, 2017 – April 30, 2018 Brendan Hassett, Director Mathew Borton, IT Director Jeff Brock, Associate Director Ruth Crane, Assistant Director Sigal Gottlieb, Deputy Director Elisenda Grigsby, Deputy Director Jeffrey Hoffstein, Consulting Associate Director Caroline Klivans, Associate Director Jill Pipher, Consulting Associate Director Bjorn Sandstede, Associate Director Homer Walker, Consulting Associate Director Ulrica Wilson, Associate Director for Diversity and Outreach Table of Contents Mission ....................................................................................................................................... 6 Core Programs and Events ......................................................................................................... 6 Participant Summaries by Program Type ................................................................................... 9 ICERM Funded Participants ................................................................................................................. 9 All Participants (ICERM funded and Non-ICERM funded) .................................................................. 10 ICERM Funded Speakers ................................................................................................................... 11 All Speakers (ICERM funded and Non-ICERM funded) ...................................................................... 12 ICERM Funded Postdocs .................................................................................................................. -
ANNOUNCEMENT of WINNERS of the Seventy-Ninth Competition Held on December 1, 2018
ANNOUNCEMENT OF WINNERS of the seventy-ninth competition held on December 1, 2018 WINNING TEAMS Rank School Team Members (in alphabetical order) 1 HARVARD UNIVERSITY Dongryul Kim, Shyam Narayanan, David Stoner 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Junyao Peng, Ashwin Sah, Yunkun Zhou 3 UNIVERSITY OF CALIFORNIA, LOS ANGELES Ciprian Mircea Bonciocat, Xiaoyu Huang, Konstantin Miagkov 4 COLUMBIA UNIVERSITY Quang Dao, Myeonhu Kim, Matthew Lerner-Brecher 5 STANFORD UNIVERSITY David Kewei Lin, Hanzhi Zheng, Yifan Zhu The institution with the first-place team receives an award of $25,000, and each member of the team receives $1,000. The awards for second place are $20,000 and $800; for third place, $15,000 and $600; for fourth place, $10,000 and $400; and for fifth place, $5,000 and $200. In each of the following categories, the listing is in alphabetical order. THE PUTNAM FELLOWS — THE FIVE HIGHEST RANKING INDIVIDUALS Each receives an award of $2,500. DONGRYUL KIM Harvard University SHYAM NARAYANAN Harvard University DAVID STONER Harvard University YUAN YAO Massachusetts Institute of Technology SHENGTONG ZHANG Massachusetts Institute of Technology THE NEXT TEN HIGHEST RANKING INDIVIDUALS Each receives an award of $1,000. MURILO CORATO ZANARELLA Princeton University JIYANG GAO Massachusetts Institute of Technology ANDREW GU Massachusetts Institute of Technology JAMES LIN Massachusetts Institute of Technology MICHAEL MA Massachusetts Institute of Technology ASHWIN SAH Massachusetts Institute of Technology KEVIN SUN Massachusetts Institute of Technology DANIELLE WANG Massachusetts Institute of Technology HUNG-HSUN YU Massachusetts Institute of Technology YUNKUN ZHOU Massachusetts Institute of Technology THE WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION ELIZABETH LOWELL PUTNAM PRIZE The winner receives an award of $1,000. -
2009 Mathematics Newsletter
AUTUMN 2009 NEWSLETTER OF THE DEPARTMENT OF MATHEMATICS AT THE UNIVERSITY OF WASHINGTON Mathematics NEWS 1 DEPARTMENT OF MATHEMATICS NEWS MESSAGE FROM THE CHAIR As you might imagine, 2009 brought Department’s work continued unabated. We have a string of unusual challenges to our department. good news to share with you in this newsletter, such as the There were uncertainties well into the UW Sophomore Medal awarded to Chad Klumb, the new- spring regarding our 2009-10 aca- found success of our students in the Putnam Competition, demic year budget. Graduate student the election of Gunther Uhlmann to the American Academy recruitment, which takes place across of Arts and Sciences, and the NSF CAREER award to Ioana the nation during the winter, was Dumitriu. Our undergraduate degree programs set new highs conducted without knowing the level at the end of the 2008-09 academic year with 368 Math of our TA funding. Similar scenarios majors, and with a total of 537 majors in Math and ACMS. played out nationally as few educational institutions were Thirteen graduate students completed the PhD, continuing spared by the financial crisis. In a survey conducted by the our recent trend of awarding significantly more PhDs than American Mathematical Society, mathematics departments the Department’s historical annual average of 6.5. reported about 900 expected faculty openings nationally, During the spring and summer, we received a boost of down from about 1,500 in a typical year, despite expecting federal funding, thanks to the excellent research projects to award 1,300 PhDs in 2009 as in recent years. -
Tobias Lee Johnson College of Staten Island, 1S-225 [email protected] 2800 Victory Blvd
Tobias Lee Johnson College of Staten Island, 1S-225 [email protected] 2800 Victory Blvd. http://www.math.csi.cuny.edu/~tobiasljohnson/ Staten Island, NY 10314 EMPLOYMENT AND EDUCATION Assistant Professor, College of Staten Island (CUNY) Fall 2017– NSF Postdoctoral Fellow New York University; sponsored by Gérard Ben Arous Fall 2016–Spring 2017 University of Southern California; sponsored by Larry Goldstein Fall 2014–Spring 2016 Ph.D. in Mathematics, University of Washington Fall 2008–Spring 2014 Advised by Ioana Dumitriu and Soumik Pal B.A. in Mathematics, Yale University Fall 2003–Spring 2007 RESEARCH Interests: probability theory and combinatorics, with a focus on discrete random structures and interacting particle systems; Stein’s method Papers: 20. Diffusion-limited annihilating systems and the increasing convex order, with Riti Bahl, Philip Barnet, and Matthew Junge. Submitted. arXiv:2104.12797 19. Particle density in diffusion-limited annihilating systems, with Matthew Junge, Hanbaek Lyu, and David Sivakoff. Preprint. arXiv:2005.06018 18. Continuous phase transitions on Galton–Watson trees. Submitted. arXiv:2007.13864 2020 17. Random tree recursions: which fixed points correspond to tangible sets of trees?, with Moumanti Podder and Fiona Skerman. Random Structures Algorithms, 56(3):796–837, 2020. arXiv:1808.03019 2019 16. Cover time for the frog model on trees, with Christopher Hoffman and Matthew Junge. Forum Math. Sigma, 7, e41 1–49, 2019. arXiv:1802.03428 15. Infection spread for the frog model on trees, with Christopher Hoffman and Matthew Junge. Electron. J. Probab., 24 (2019), no. 112, 1–29. arXiv:1710.05884 14. Sensitivity of the frog model to initial conditions, with Leonardo T. -
On the Second Eigenvalue of Random Bipartite Biregular Graphs
ON THE SECOND EIGENVALUE OF RANDOM BIPARTITE BIREGULAR GRAPHS YIZHE ZHU Abstract. We consider the spectral gap of a uniformly chosen random (d1; d2)-biregular bipartite graph G with jV j = n; jV j = m, where d ; d could possibly grow with n and m. Let A be the adjacency matrix 1 2 1 2 p 2=3 of G. Under the assumption that d1 ≥ d2 and d2 = O(n ); we show that λ2(A) = O( d1) with high probability. As a corollary, combining the results from [47],p we confirm a conjecture in [14] that the second singular value of a uniform random d-regular digraph is O( d) for 1 ≤ d ≤ n=2 with highp probability. This also implies that the second eigenvalue of a uniform random d-regular digraph is O( d) for 1 ≤ d ≤ n=2 with high probability. 2 Assuming d2 = O(1) and d1 = O(n ), we further prove that for a random (d1; d2)-biregular bipartite 2 p graph, jλi (A) − d1j = O( d1(d2 − 1)) for all 2 ≤ i ≤ n + m − 1 with high probability. The proofs of the two results are based on the size biased coupling method introduced in [13] for random d-regular graphs and several new switching operations we defined for random bipartite biregular graphs. 1. Introduction An expander graph is a sparse graph that has strong connectivity properties and exhibits rapid mixing. Expander graphs play an important role in computer science including sampling, complexity theory, and the design of error-correcting codes (see [27]). When a graph is d-regular, i.e., each vertex has degree d, quantification of expansion is possible based on the eigenvalues of the adjacency matrix. -
Proposed MS in Data Science at Its October 1
ACADEMIC SENATE: SAN DIEGO DIVISION, 0002 UCSD, LA JOLLA, CA 92093-0002 (858) 534-3640 FAX (858) 534-4528 October 27, 2020 PROFESSOR RAJESH GUPTA Halicioğlu Data Science Institute SUBJECT: Proposed MS in Data Science At its October 12, 2020 meeting, the Graduate Council approved the proposal to establish a graduate degree program of study leading to a Master of Science (M.S.) in Data Science. The Graduate Council will forward the proposal for placement on an upcoming Representative Assembly agenda. Please note that proposers may not accept applications to the program or admit students until systemwide review of the proposal is complete and the UC Office of the President has issued a final outcome. Sincerely, Lynn Russell, Chair Graduate Council cc: M. Allen J. Antony S. Constable R. Continetti B. Cowan T. Javidi C. Lyons J. Morgan K. Ng R. Rodriguez D. Salmon Y. Wo l l ma n A Proposal for a Program of Graduate Study in Data Science leading to a degree in Master of Science in Data Science (MS/DS) By: MS Program Committee, HDSI Yoav Freund, Shankar Subramaniam, Virginia de Sa, Ery Arias-Castro, Armin Schwartzman, Dimitris Politis, Ilkay Altintas, Rayan Saab. Contacts: Academic: Rajesh K. Gupta Director, Halıcıoğlu Data Science Institute (HDSI) (858) 822-4391 [email protected] Administrative: Yvonne Wollman Student Affairs Manager Halıcıoğlu Data Science Institute (HDSI) (858) 246-5427 [email protected] Version History: March 6, 2020: Version 1.0 submitted for review by graduate division. April 12, 2020: Version 2.0 submitted to academic senate and revised administrative review. May 22, 2020: Version 3.0 submitted to academic senate after administrative revisions. -
Gabriel Carroll Curriculum Vitae Updated April 21, 2021 Department
Gabriel Carroll Curriculum Vitae Updated April 21, 2021 Department of Economics, University of Toronto Max Gluskin House, 150 St. George Street, Toronto, Ontario M5S 3G7, Canada http://individual.utoronto.ca/carroll j [email protected] Personal Information Birthdate: December 24, 1982 Citizenship: USA Education PhD in Economics, Massachusetts Institute of Technology, 2012 Doctoral thesis: \Approaches to Mechanism Design with Boundedly Rational Agents" Advisors: Parag Pathak, Daron Acemoglu, Glenn Ellison BA in Mathematics and Linguistics, summa cum laude, Harvard College, 2005 Employment Associate professor (tenured) in Department of Economics, University of Toronto, January 2021 { present Associate professor (tenured) in Department of Economics, Stanford University, July 2020 { December 2020 Assistant professor in Department of Economics, Stanford University, July 2013 { June 2020 Post-doctoral researcher at Microsoft Research, July 2012 { June 2013 Instructor at Mathematical Olympiad Summer Program, Summer 2008, 2009, 2010, 2012 Research assistant for James Choi, David Laibson, and Brigitte Madrian at NBER, August 2006 { July 2007 Volunteer English teacher at Chaling #1 Middle School, Chaling, Hunan, China, August 2005 { July 2006 Visiting Positions (one month or more) Visitor at Research School of Economics, Australian National University, March { May 2018; 1 March { May 2019 Visitor at Cowles Foundation, Yale University, February { March 2018 Part-time visiting scholar at Simons Institute, UC Berkeley, August { December