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2006 Annual Report
Contents Clay Mathematics Institute 2006 James A. Carlson Letter from the President 2 Recognizing Achievement Fields Medal Winner Terence Tao 3 Persi Diaconis Mathematics & Magic Tricks 4 Annual Meeting Clay Lectures at Cambridge University 6 Researchers, Workshops & Conferences Summary of 2006 Research Activities 8 Profile Interview with Research Fellow Ben Green 10 Davar Khoshnevisan Normal Numbers are Normal 15 Feature Article CMI—Göttingen Library Project: 16 Eugene Chislenko The Felix Klein Protocols Digitized The Klein Protokolle 18 Summer School Arithmetic Geometry at the Mathematisches Institut, Göttingen, Germany 22 Program Overview The Ross Program at Ohio State University 24 PROMYS at Boston University Institute News Awards & Honors 26 Deadlines Nominations, Proposals and Applications 32 Publications Selected Articles by Research Fellows 33 Books & Videos Activities 2007 Institute Calendar 36 2006 Another major change this year concerns the editorial board for the Clay Mathematics Institute Monograph Series, published jointly with the American Mathematical Society. Simon Donaldson and Andrew Wiles will serve as editors-in-chief, while I will serve as managing editor. Associate editors are Brian Conrad, Ingrid Daubechies, Charles Fefferman, János Kollár, Andrei Okounkov, David Morrison, Cliff Taubes, Peter Ozsváth, and Karen Smith. The Monograph Series publishes Letter from the president selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. The next volume in the series will be Ricci Flow and the Poincaré Conjecture, by John Morgan and Gang Tian. Their book will appear in the summer of 2007. In related publishing news, the Institute has had the complete record of the Göttingen seminars of Felix Klein, 1872–1912, digitized and made available on James Carlson. -
Manjul Bhargava
The Work of Manjul Bhargava Manjul Bhargava's work in number theory has had a profound influence on the field. A mathematician of extraordinary creativity, he has a taste for simple problems of timeless beauty, which he has solved by developing elegant and powerful new methods that offer deep insights. When he was a graduate student, Bhargava read the monumental Disqui- sitiones Arithmeticae, a book about number theory by Carl Friedrich Gauss (1777-1855). All mathematicians know of the Disquisitiones, but few have actually read it, as its notation and computational nature make it difficult for modern readers to follow. Bhargava nevertheless found the book to be a wellspring of inspiration. Gauss was interested in binary quadratic forms, which are polynomials ax2 +bxy +cy2, where a, b, and c are integers. In the Disquisitiones, Gauss developed his ingenious composition law, which gives a method for composing two binary quadratic forms to obtain a third one. This law became, and remains, a central tool in algebraic number theory. After wading through the 20 pages of Gauss's calculations culminating in the composition law, Bhargava knew there had to be a better way. Then one day, while playing with a Rubik's cube, he found it. Bhargava thought about labeling each corner of a cube with a number and then slic- ing the cube to obtain 2 sets of 4 numbers. Each 4-number set naturally forms a matrix. A simple calculation with these matrices resulted in a bi- nary quadratic form. From the three ways of slicing the cube, three binary quadratic forms emerged. -
Orbital Resonances and GPU Acceleration of Binary Black Hole Inspiral Simulations
Orbital resonances and GPU acceleration of binary black hole inspiral simulations by Adam Gabriel Marcel Lewis A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Physics University of Toronto c Copyright 2018 by Adam Gabriel Marcel Lewis Abstract Orbital resonances and GPU acceleration of binary black hole inspiral simulations Adam Gabriel Marcel Lewis Doctor of Philosophy Department of Physics University of Toronto 2018 Numerical relativity, the direct numerical integration of the Einstein field equations, is now a mature subfield of computational physics, playing a critical role in the generation of signal templates for comparison with data from ground-based gravitational wave de- tectors. The application of numerical relativity techniques to new problems is at present complicated by long wallclock times and intricate code. In this thesis we lay groundwork to improve this situation by presenting a GPU port of the numerical relativity code SpEC. Our port keeps code maintenance feasible by relying on various layers of automation, and achieves high performance across a variety of GPUs. We secondly introduce a C++ soft- ware package, TLoops, which allows numerical manipulation of tensors using single-line C++ source-code expressions resembling familiar tensor calculus notation. These expres- sions may be compiled and executed immediately, but also can be used to automatically generate equivalent GPU or low-level CPU code, which then executes in their place. The GPU code in particular achieves near-peak performance. Finally, we present simulations of eccentric binary black holes. We develop new methods to extract the fundamental fre- quencies of these systems. -
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. -
Comments on the 2011 Shaw Prize in Mathematical Sciences - - an Analysis of Collectively Formed Errors in Physics by C
Global Journal of Science Frontier Research Physics and Space Science Volume 12 Issue 4 Version 1.0 June 2012 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Comments on the 2011 Shaw Prize in Mathematical Sciences - - An Analysis of Collectively Formed Errors in Physics By C. Y. Lo Applied and Pure Research Institute, Nashua, NH Abstract - The 2011 Shaw Prize in mathematical sciences is shared by Richard S. Hamilton and D. Christodoulou. However, the work of Christodoulou on general relativity is based on obscure errors that implicitly assumed essentially what is to be proved, and thus gives misleading results. The problem of Einstein’s equation was discovered by Gullstrand of the 1921 Nobel Committee. In 1955, Gullstrand is proven correct. The fundamental errors of Christodoulou were due to his failure to distinguish the difference between mathematics and physics. His subsequent errors in mathematics and physics were accepted since judgments were based not on scientific evidence as Galileo advocates, but on earlier incorrect speculations. Nevertheless, the Committee for the Nobel Prize in Physics was also misled as shown in their 1993 press release. Here, his errors are identified as related to accumulated mistakes in the field, and are illustrated with examples understandable at the undergraduate level. Another main problem is that many theorists failed to understand the principle of causality adequately. It is unprecedented to award a prize for mathematical errors. Keywords : Nobel Prize; general relativity; Einstein equation, Riemannian Space; the non- existence of dynamic solution; Galileo. GJSFR-A Classification : 04.20.-q, 04.20.Cv Comments on the 2011 Shaw Prize in Mathematical Sciences -- An Analysis of Collectively Formed Errors in Physics Strictly as per the compliance and regulations of : © 2012. -
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Arxiv:1407.1093V1 [Math.NT]
MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER Abstract. We show that the cyclotomic Iwasawa–Greenberg Main Conjecture holds for a large class of modular forms with multiplicative reduction at p, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument. 1. Introduction The cyclotomic Iwasawa–Greenberg Main Conjecture was established in [18], in com- bination with work of Kato [13], for a large class of newforms f ∈ Sk(Γ0(N)) that are ordinary at an odd prime p ∤ N, subject to k ≡ 2 (mod p − 1) and certain conditions on the mod p Galois representation associated with f. The purpose of this note is to extend this result to the case where p | N (in which case k is necessarily equal to 2). ∞ n Recall that the coefficients an of the q-expansion f = n=1 anq of f at the cusp at infinity (equivalently, the Hecke eigenvalues of f) are algebraic integers that generate a finite extension Q(f) ⊂ C of Q. Let p be an odd primeP and let L be a finite extension of the completion of Q(f) at a chosen prime above p (equivalently, let L be a finite extension of Qp in a fixed algebraic closure Qp of Qp that contains the image of a chosen embedding Q(f) ֒→ Qp). Suppose that f is ordinary at p with respect to L in the sense that ap is a unit in the ring of integers O of L. -
New Publications Offered by The
New Publications Offered by the AMS To subscribe to email notification of new AMS publications, please go to www.ams.org/bookstore-email. Algebra and Algebraic Geometry theorem in noncommutative geometry; M. Guillemard, An overview of groupoid crossed products in dynamical systems. Contemporary Mathematics, Volume 676 Noncommutative November 2016, 222 pages, Softcover, ISBN: 978-1-4704-2297-4, LC 2016017993, 2010 Mathematics Subject Classification: 00B25, 46L87, Geometry and Optimal 58B34, 53C17, 46L60, AMS members US$86.40, List US$108, Order Transport code CONM/676 Pierre Martinetti, Università di Genova, Italy, and Jean-Christophe Wallet, CNRS, Université Paris-Sud 11, Orsay, France, Editors Differential Equations This volume contains the proceedings of the Workshop on Noncommutative Geometry and Optimal Transport, held on November 27, 2014, in Shock Formation in Besançon, France. The distance formula in noncommutative geometry was introduced Small-Data Solutions to by Connes at the end of the 1980s. It is a generalization of Riemannian 3D Quasilinear Wave geodesic distance that makes sense in a noncommutative setting, and provides an original tool to study the geometry of the space of Equations states on an algebra. It also has an intriguing echo in physics, for it yields a metric interpretation for the Higgs field. In the 1990s, Jared Speck, Massachusetts Rieffel noticed that this distance is a noncommutative version of the Institute of Technology, Cambridge, Wasserstein distance of order 1 in the theory of optimal transport. MA More exactly, this is a noncommutative generalization of Kantorovich dual formula of the Wasserstein distance. Connes distance thus offers In 1848 James Challis showed that an unexpected connection between an ancient mathematical problem smooth solutions to the compressible and the most recent discovery in high energy physics. -
A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol
Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members. -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
Complex Analysis (Princeton Lectures in Analysis, Volume
COMPLEX ANALYSIS Ibookroot October 20, 2007 Princeton Lectures in Analysis I Fourier Analysis: An Introduction II Complex Analysis III Real Analysis: Measure Theory, Integration, and Hilbert Spaces Princeton Lectures in Analysis II COMPLEX ANALYSIS Elias M. Stein & Rami Shakarchi PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright © 2003 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Control Number 2005274996 ISBN 978-0-691-11385-2 British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 5 7 9 10 8 6 To my grandchildren Carolyn, Alison, Jason E.M.S. To my parents Mohamed & Mireille and my brother Karim R.S. Foreword Beginning in the spring of 2000, a series of four one-semester courses were taught at Princeton University whose purpose was to present, in an integrated manner, the core areas of analysis. The objective was to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. The present series of books is an elaboration of the lectures that were given. While there are a number of excellent texts dealing with individual parts of what we cover, our exposition aims at a different goal: pre- senting the various sub-areas of analysis not as separate disciplines, but rather as highly interconnected. -
Probabilistic Models on Fibre Bundles by Shan Shan
Probabilistic Models on Fibre Bundles by Shan Shan Department of Mathematics Duke University Date: Approved: Ingrid Daubechies, Supervisor Sayan Mukherjee, Chair Doug Boyer Colleen Robles Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Probabilistic Models on Fibre Bundles by Shan Shan Department of Mathematics Duke University Date: Approved: Ingrid Daubechies, Supervisor Sayan Mukherjee, Chair Doug Boyer Colleen Robles An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Shan Shan All rights reserved Abstract In this thesis, we propose probabilistic models on fibre bundles for learning the gen- erative process of data. The main tool we use is the diffusion kernel and we use it in two ways. First, we build from the diffusion kernel on a fibre bundle a projected kernel that generates robust representations of the data, and we test that it outperforms regular diffusion maps under noise. Second, this diffusion kernel gives rise to a nat- ural covariance function when defining Gaussian processes (GP) on the fibre bundle. To demonstrate the uses of GP on a fibre bundle, we apply it to simulated data on a M¨obiusstrip for the problem of prediction and regression. Parameter tuning can also be guided by a novel semi-group test arising from the geometric properties of dif- fusion kernel. For an example of real-world application, we use probabilistic models on fibre bundles to study evolutionary process on anatomical surfaces. -
2. Superoscillations.Pdf
When the Whole Vibrates Faster than Any of its Parts Computational Superoscillation Theory Nicholas Wheeler December 2017 Introduction. The phenomenon/theory of “superoscillations”—brought to my attention by my friend Ahmed Sebbar1—originated in the time-symmetric formulation of quantum mechanics2 that gave rise to the theory of “weak measurements.” Recognition of the role that superoscillations play in that work was brought into explicit focus by Aharonov et al in 1990, and since that date the concept has found application to a remarkable variety of physical subject areas. Yakir Aharonov took his PhD in 1960 from the University of Bristol, where he worked with David Bohm.3 In 1992 a conference was convened at Bristol to celebrate Aharonov’s 60th birthday. It was on that occasion that the superoscillation phenomenon came first to the attention of Michael Berry (of the Bristol faculty), who in an essay “Faster than Fourier” contributed to the proceedings of that conference4 wrote that “He [Aharonov] told me that it is possible for functions to oscillate faster than any of their Fourier components. This seemed unbelievable, even paradoxical; I had heard nothing like it before. ” Berry was inspired to write a series of papers relating to the theory of superoscillation and its potential applications, as also by now have a great many other authors. 1 Private communication, 17 November 2017. 2 Y. Aharonov, P. G. Bergmann & J. L. Libowitz, “Time symmetry in the quantum meassurement process,” Phys. Rev. B 134, 1410–1416 (1964). 3 It was, I suspect, at the invitation of E. P Gross—who had collaborated with Bohm when both were at Princeton—that Aharonov spent 1960–1961 at Brandeis University, from which I had taken the first PhD and departed to Utrecht/CERN in February of 1960, so we never met.