MATHEMATICS from the Chair

Total Page:16

File Type:pdf, Size:1020Kb

MATHEMATICS from the Chair 2020–2021 • Issue 9 Department of MATHEMATICS Princeton University From the Chair Teaching During a Pandemic I would like to extend my sin- "The first indication we had of the bining live lectures and office hours cerest congratulations to the class coming disruption was at the very with pre-recorded content. Instructor of 2021, as well as those graduate beginning of the spring 2020 semester Dr. Andrew Yarmola taught MAT201: students who will complete their when junior faculty who'd spent the “The lecture itself contained basic final public orals over the next few winter abroad were having difficulty examples, the pre-recorded problems months. While the past two years getting back into the country. were usually more detailed and al- have presented some unusual ob- "Big restrictions kicked in suddenly lowed students to focus on specific stacles, I am proud to see our stu- during midterm week. At that point topics (and review as they saw fit). dents continue to thrive academically everyone was trying to be hopeful All-in-all, my sense is that the students as they rally towards their goals. I am also happy to welcome the that life would return to normal in saw more worked-out problems in live undergraduate classes of 2019 and a few weeks, but it rapidly became Zoom precepts and pre-recorded prob- 2020, as well as those graduate clear that optimism might not be the lems vs. a usual non-virtual year.” students who have completed their best strategy." Dr. Jennifer Johnson, A variety of approaches gave us a PhDs since our last publication, as Senior Lecturer and Associate Direc- better idea of what was and was not new members of our distinguished tor of Undergraduate Studies oversees working. “We saw through student alumni. the department’s large, multi-section feedback that higher production videos Much has changed on a global calculus and linear algebra courses, did not necessarily equate to a better scale in recent times, and the Princ- and as it became clear that the spring learning experience for the students; eton University Math Department term would have to transition to an they much preferred videos that felt holds no exemption. But while we online format she began work with our more ‘real’ to them, for example videos find ourselves in a time of unique dedicated faculty to ensure the spring recorded at a chalkboard” recalls Dr. challenges, it is also proving to be semester was able to carry on and to John Fickenscher, Associate Research a time of exceptional achievements. prepare for the likelihood of a full year Scholar and Lecturer. Last spring, when the university transitioned to online teaching and of virtual teaching. While recorded course materials research in response to the COV- Going Virtual worked well as supplementary mate- ID-19 pandemic, we found ourselves The immediate move online in the rial to the courses, Dr. Johnson saw in largely unfamiliar territory. De- spring of 2020 naturally led to experi- in mid-semester feedback that stu- spite this, I am pleased to report that mentation with teaching styles, com- ...continued on page 11 the math department was able to le- verage academic, administrative, and technological resources to swiftly or- Inside this issue: chestrate the framework for this new Faculty Retirements— Page 4 way of learning, which continued to Faculty Memorials— Page 5–6 improve over the 2020-2021 academ- ic year. In addition to the facilitation Honors and Awards — Pages 6–7 of online classes, the department has Climate and Inclusion — Page 9 also kept its seminar program active Alumni News — Page 9 with an online schedule that matches that of previous years, and continues Special Events — Page 10 ...continued on next page Undergraduate Program — Page 13 Graduate Program — Page 14 From the Chair from the Chair ...continued from previous page tures. This July, Yakov Sinai will also to cultivate academic collaboration as The department also created a new retire to Emeritus status after twenty- we adapt to these unexpected changes. Climate and Inclusion Committee eight years with the department. It is inspiring to witness the unwaver- this year. Chaired by Professor Peter Among his many professional achieve- ing perseverance, enthusiasm, and ca- Sarnak, this committee of faculty, ments, Sinai's extensive work in math- maraderie of our faculty, students, and students, alumni, and staff has begun ematical physics and probability theory staff as they rise to meet the challenges the hard work of addressing important earned him the Abel Prize in 2014. of such an unprecedented chapter in issues to our community. In addition He has also made a significant impact our university’s history. to town hall meetings and directed as a teacher, and has advised over 50 This year, we welcomed one new mentorship programs, a new Horizons students throughout his career. Assistant Professor, four new instruc- Lectures series began this year, where Members of the department have tors, and four new researchers to the speakers are invited to give both a continued to receive accolades and department. We are also pleased to research talk and host a seminar on awards in the field of mathematics. report that seven new instructors and the societal impacts of mathematics To highlight a few of many notable assistant professors have accepted with respect to diversity, inclusion, and achievements, Aleksandr Logunov was offers to join our junior faculty over mentorship. Our inaugural Horizons named both a Packard Fellow and a the next couple of years. During this Lecturer was Trachette Jackson of the Sloan Research Fellow, and received academic year, we were honored to University of Michigan, who hosted a the New Horizons Prize; and Tristan have Chenyang Xu join our depart- compelling panel discussion. Buckmaster received a Clay Research ment as a Professor of Mathematics. Since the last publication of Fine Award (joint with Philip Isett *13 and We welcomed Jacob Rasmussen of Letters, the department lost two stellar former Assistant Professor Vlad Vi- the University of Cambridge, as our mathematicians. Professor Emeritus col). Our alumni have also continued Minerva Distinguished Visitor in the Goro Shimura, internationally re- a tradition of success, with Hillel Fur- spring of 2020, and James Maynard of nowned within the fields of modern stenberg *58 receiving the Abel Prize, the University of Oxford delivered an number theory, arithmetic geometry and Alex Eskin *93 taking home the online series of three Minerva Lec- and automorphic forms, passed away Breakthrough Prize. in May of 2019. We also lost Professor In closing, allow me to express my Department Administration Emeritus John Conway, who passed disappointment that alumni weekend Chair last April. A prolific and imaginative events, and many end-of-year activi- Igor Rodnianski thinker, Conway’s enormous math- ties, are once again canceled as a result ematical contributions will leave a of the pandemic. The department’s Associate Chair lasting mark on current and future Alumni Reception is an important János Kollár generations of mathematicians. tradition, and I count myself among Departmental Representative At the end of the 2019-2020 aca- the many members of the department János Kollár demic year, we celebrated the retire- who look forward to reconnecting with ment of Professor Robert Gunning, alumni during this exciting time of Director of Graduate Studies Mihalis Dafermos who has been a member of the depart- year. Despite this set back, I know that Evita Nestoridi ment since 1956. Among his many many of you will continue to stay in contributions to the university, Robert touch with members of the Fine Hall Senior Advisors served as Chair of the Mathemat- family, and I look forward to welcom- John Pardon ics Department, Chief Marshall for ing you all at future events. Junior Advisor University Convocations, and Dean of Mark McConnell the Faculty. An extraordinary teacher, he also received the President’s Award Placement Officer for Distinguished Teaching in 2003. Ana Menezes He transitioned to Emeritus status last Igor Rodnianski year. Department Chair Page Faculty Appointments Professor Chenyang Xu 2019-20 2020-21 The depart- Academic Year Academic Year ment is Daniel Álvarez Gavela Allen David Boozer pleased to Instructor Instructor Topology announce the Symplectic Geometry appointment Jacob Carruth Alan Chang of Chenyang Postdoctoral Research Associate Instructor Xu as a Profes- Analysis Analysis sor of Math- Duncan Dauvergne ematics. Instructor Evra Shai Instructor A leader Probability Number Theory in Algebraic Michele Fornea Geometry with a focus on birational Postdoctoral Research Associate Tristan Leger geometry, Professor Xu completed his Number Theory Postdoctoral Research Associate Analysis PhD here at Princeton in 2008 under Yusuf Bariş Kartal the supervision of Professor János Kol- Instructor lár. Xu was a CLE Moore Instructor at Symplectic Topology Sarah Peluse Veblen Research Instructor MIT from 2008-2011, after which he Ben Krause Combinatorics, Number Theory took positions as an assistant profes- Postdoctoral Research Associate sor at the University of Utah beforing Analysis Hannah Schwartz Postdoctoral Research Associate joining Peking University as a research Nicholas Marshall fellow and then a professor. Most re- Postdoctoral Research Fellow Topology cently Xu was a professor at MIT. Harmonic Analysis John Sheridan Professor Xu has received nu- Joaquín Moraga Postdoctoral Research Associate merous awards, including the 2016 Instructor Algebraic
Recommended publications
  • MY UNFORGETTABLE EARLY YEARS at the INSTITUTE Enstitüde Unutulmaz Erken Yıllarım
    MY UNFORGETTABLE EARLY YEARS AT THE INSTITUTE Enstitüde Unutulmaz Erken Yıllarım Dinakar Ramakrishnan `And what was it like,’ I asked him, `meeting Eliot?’ `When he looked at you,’ he said, `it was like standing on a quay, watching the prow of the Queen Mary come towards you, very slowly.’ – from `Stern’ by Seamus Heaney in memory of Ted Hughes, about the time he met T.S.Eliot It was a fortunate stroke of serendipity for me to have been at the Institute for Advanced Study in Princeton, twice during the nineteen eighties, first as a Post-doctoral member in 1982-83, and later as a Sloan Fellow in the Fall of 1986. I had the privilege of getting to know Robert Langlands at that time, and, needless to say, he has had a larger than life influence on me. It wasn’t like two ships passing in the night, but more like a rowboat feeling the waves of an oncoming ship. Langlands and I did not have many conversations, but each time we did, he would make a Zen like remark which took me a long time, at times months (or even years), to comprehend. Once or twice it even looked like he was commenting not on the question I posed, but on a tangential one; however, after much reflection, it became apparent that what he had said had an interesting bearing on what I had been wondering about, and it always provided a new take, at least to me, on the matter. Most importantly, to a beginner in the field like I was then, he was generous to a fault, always willing, whenever asked, to explain the subtle aspects of his own work.
    [Show full text]
  • Advanced Algebra
    Cornerstones Series Editors Charles L. Epstein, University of Pennsylvania, Philadelphia Steven G. Krantz, University of Washington, St. Louis Advisory Board Anthony W. Knapp, State University of New York at Stony Brook, Emeritus Anthony W. Knapp Basic Algebra Along with a companion volume Advanced Algebra Birkhauser¨ Boston • Basel • Berlin Anthony W. Knapp 81 Upper Sheep Pasture Road East Setauket, NY 11733-1729 U.S.A. e-mail to: [email protected] http://www.math.sunysb.edu/˜ aknapp/books/b-alg.html Cover design by Mary Burgess. Mathematics Subject Classicification (2000): 15-01, 20-02, 13-01, 12-01, 16-01, 08-01, 18A05, 68P30 Library of Congress Control Number: 2006932456 ISBN-10 0-8176-3248-4 eISBN-10 0-8176-4529-2 ISBN-13 978-0-8176-3248-9 eISBN-13 978-0-8176-4529-8 Advanced Algebra ISBN 0-8176-4522-5 Basic Algebra and Advanced Algebra (Set) ISBN 0-8176-4533-0 Printed on acid-free paper. c 2006 Anthony W. Knapp All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
    [Show full text]
  • Quadratic Forms and Automorphic Forms
    Quadratic Forms and Automorphic Forms Jonathan Hanke May 16, 2011 2 Contents 1 Background on Quadratic Forms 11 1.1 Notation and Conventions . 11 1.2 Definitions of Quadratic Forms . 11 1.3 Equivalence of Quadratic Forms . 13 1.4 Direct Sums and Scaling . 13 1.5 The Geometry of Quadratic Spaces . 14 1.6 Quadratic Forms over Local Fields . 16 1.7 The Geometry of Quadratic Lattices – Dual Lattices . 18 1.8 Quadratic Forms over Local (p-adic) Rings of Integers . 19 1.9 Local-Global Results for Quadratic forms . 20 1.10 The Neighbor Method . 22 1.10.1 Constructing p-neighbors . 22 2 Theta functions 25 2.1 Definitions and convergence . 25 2.2 Symmetries of the theta function . 26 2.3 Modular Forms . 28 2.4 Asymptotic Statements about rQ(m) ...................... 31 2.5 The circle method and Siegel’s Formula . 32 2.6 Mass Formulas . 34 2.7 An Example: The sum of 4 squares . 35 2.7.1 Canonical measures for local densities . 36 2.7.2 Computing β1(m) ............................ 36 2.7.3 Understanding βp(m) by counting . 37 2.7.4 Computing βp(m) for all primes p ................... 38 2.7.5 Computing rQ(m) for certain m ..................... 39 3 Quaternions and Clifford Algebras 41 3.1 Definitions . 41 3.2 The Clifford Algebra . 45 3 4 CONTENTS 3.3 Connecting algebra and geometry in the orthogonal group . 47 3.4 The Spin Group . 49 3.5 Spinor Equivalence . 52 4 The Theta Lifting 55 4.1 Classical to Adelic modular forms for GL2 ..................
    [Show full text]
  • For Your Information
    fyi.qxp 3/18/98 3:19 PM Page 244 For Your Information structure federal support. The study is part of a larger ef- International Study of fort at the Academy to gauge where the U.S. stands in- Mathematics and Science ternationally in scientific research. The motivation comes from a 1993 report by the Committee on Science, Engi- Achievement neering, and Public Policy (COSEPUP), which set forth strate- gies for making decisions about how best to use federal The first installment of results from the Third International research funds. COSEPUP is a committee of the NAS, the Mathematics and Science Study (TIMSS) was released on No- National Academy of Engineering, and the Institute of Med- vember 20. This first batch of data pertains to achievement icine. The COSEPUP report recommended that the U.S. aim of eighth-graders; later releases will focus on fourth- and to be the world leader in certain critical fields and to be twelfth-graders. The study found that U.S. eighth-graders among the leaders in other areas. The report urged field- performed below the international average in mathemat- by-field assessments by independent panels of researchers ics but slightly above average in science. The U.S. was in the field, researchers in closely related fields, and users among thirty-three countries in which there was no sta- of the research. The mathematical sciences study is the first tistically significant difference between the performance such assessment. If this project is successful, the Academy of eighth-grade boys and girls in mathematics. The study will follow suit with other areas.
    [Show full text]
  • A Unified Approach to Three Themes in Harmonic Analysis ($1^{St} $ Part)
    A UNIFIED APPROACH TO THREE THEMES IN HARMONIC ANALYSIS (I & II) (I) THE LINEAR HILBERT TRANSFORM AND MAXIMAL OPERATOR ALONG VARIABLE CURVES (II) CARLESON TYPE OPERATORS IN THE PRESENCE OF CURVATURE (III) THE BILINEAR HILBERT TRANSFORM AND MAXIMAL OPERATOR ALONG VARIABLE CURVES VICTOR LIE In loving memory of Elias Stein, whose deep mathematical breadth and vision for unified theories and fundamental concepts have shaped the field of harmonic analysis for more than half a century. arXiv:1902.03807v2 [math.AP] 2 Oct 2020 Date: October 5, 2020. Key words and phrases. Wave-packet analysis, Hilbert transform and Maximal oper- ator along curves, Carleson-type operators in the presence of curvature, bilinear Hilbert transform and maximal operators along curves, Zygmund’s differentiation conjecture, Car- leson’s Theorem, shifted square functions, almost orthogonality. The author was supported by the National Science Foundation under Grant No. DMS- 1500958. The most recent revision of the paper was performed while the author was supported by the National Science Foundation under Grant No. DMS-1900801. 1 2 VICTOR LIE Abstract. In the present paper and its sequel [63], we address three rich historical themes in harmonic analysis that rely fundamentally on the concept of non-zero curvature. Namely, we focus on the boundedness properties of (I) the linear Hilbert transform and maximal operator along variable curves, (II) Carleson-type operators in the presence of curvature, and (III) the bilinear Hilbert transform and maximal operator along variable curves. Our Main Theorem states that, given a general variable curve γ(x,t) in the plane that is assumed only to be measurable in x and to satisfy suitable non-zero curvature (in t) and non-degeneracy conditions, all of the above itemized operators defined along the curve γ are Lp-bounded for 1 <p< ∞.
    [Show full text]
  • A Glimpse of the Laureate's Work
    A glimpse of the Laureate’s work Alex Bellos Fermat’s Last Theorem – the problem that captured planets moved along their elliptical paths. By the beginning Andrew Wiles’ imagination as a boy, and that he proved of the nineteenth century, however, they were of interest three decades later – states that: for their own properties, and the subject of work by Niels Henrik Abel among others. There are no whole number solutions to the Modular forms are a much more abstract kind of equation xn + yn = zn when n is greater than 2. mathematical object. They are a certain type of mapping on a certain type of graph that exhibit an extremely high The theorem got its name because the French amateur number of symmetries. mathematician Pierre de Fermat wrote these words in Elliptic curves and modular forms had no apparent the margin of a book around 1637, together with the connection with each other. They were different fields, words: “I have a truly marvelous demonstration of this arising from different questions, studied by different people proposition which this margin is too narrow to contain.” who used different terminology and techniques. Yet in the The tantalizing suggestion of a proof was fantastic bait to 1950s two Japanese mathematicians, Yutaka Taniyama the many generations of mathematicians who tried and and Goro Shimura, had an idea that seemed to come out failed to find one. By the time Wiles was a boy Fermat’s of the blue: that on a deep level the fields were equivalent. Last Theorem had become the most famous unsolved The Japanese suggested that every elliptic curve could be problem in mathematics, and proving it was considered, associated with its own modular form, a claim known as by consensus, well beyond the reaches of available the Taniyama-Shimura conjecture, a surprising and radical conceptual tools.
    [Show full text]
  • “To Be a Good Mathematician, You Need Inner Voice” ”Огонёкъ” Met with Yakov Sinai, One of the World’S Most Renowned Mathematicians
    “To be a good mathematician, you need inner voice” ”ОгонёкЪ” met with Yakov Sinai, one of the world’s most renowned mathematicians. As the new year begins, Russia intensifies the preparations for the International Congress of Mathematicians (ICM 2022), the main mathematical event of the near future. 1966 was the last time we welcomed the crème de la crème of mathematics in Moscow. “Огонёк” met with Yakov Sinai, one of the world’s top mathematicians who spoke at ICM more than once, and found out what he thinks about order and chaos in the modern world.1 Committed to science. Yakov Sinai's close-up. One of the most eminent mathematicians of our time, Yakov Sinai has spent most of his life studying order and chaos, a research endeavor at the junction of probability theory, dynamical systems theory, and mathematical physics. Born into a family of Moscow scientists on September 21, 1935, Yakov Sinai graduated from the department of Mechanics and Mathematics (‘Mekhmat’) of Moscow State University in 1957. Andrey Kolmogorov’s most famous student, Sinai contributed to a wealth of mathematical discoveries that bear his and his teacher’s names, such as Kolmogorov-Sinai entropy, Sinai’s billiards, Sinai’s random walk, and more. From 1998 to 2002, he chaired the Fields Committee which awards one of the world’s most prestigious medals every four years. Yakov Sinai is a winner of nearly all coveted mathematical distinctions: the Abel Prize (an equivalent of the Nobel Prize for mathematicians), the Kolmogorov Medal, the Moscow Mathematical Society Award, the Boltzmann Medal, the Dirac Medal, the Dannie Heineman Prize, the Wolf Prize, the Jürgen Moser Prize, the Henri Poincaré Prize, and more.
    [Show full text]
  • CURRICULUM VITAE November 2007 Hugo J
    CURRICULUM VITAE November 2007 Hugo J. Woerdeman Professor and Department Head Office address: Home address: Department of Mathematics 362 Merion Road Drexel University Merion, PA 19066 Philadelphia, PA 19104 Phone: (610) 664-2344 Phone: (215) 895-2668 Fax: (215) 895-1582 E-mail: [email protected] Academic employment: 2005– Department of Mathematics, Drexel University Professor and Department Head (January 2005 – Present) 1989–2004 Department of Mathematics, College of William and Mary, Williamsburg, VA. Margaret L. Hamilton Professor of Mathematics (August 2003 – December 2004) Professor (July 2001 – December 2004) Associate Professor (September 1995 – July 2001) Assistant Professor (August 1989 – August 1995; on leave: ’89/90) 2002-03 Department of Mathematics, K. U. Leuven, Belgium, Visiting Professor Post-doctorate: 1989– 1990 University of California San Diego, Advisor: J. W. Helton. Education: Ph. D. degree in mathematics from Vrije Universiteit, Amsterdam, 1989. Thesis: ”Matrix and Operator Extensions”. Advisor: M. A. Kaashoek. Co-advisor: I. Gohberg. Doctoraal (equivalent of M. Sc.), Vrije Universiteit, Amsterdam, The Netherlands, 1985. Thesis: ”Resultant Operators and the Bezout Equation for Analytic Matrix Functions”. Advisor: L. Lerer Current Research Interests: Modern Analysis: Operator Theory, Matrix Analysis, Optimization, Signal and Image Processing, Control Theory, Quantum Information. Editorship: Associate Editor of SIAM Journal of Matrix Analysis and Applications. Guest Editor for a Special Issue of Linear Algebra and
    [Show full text]
  • Your Project Title
    Curriculum vitae Alfonso Sorrentino Curriculum Vitae • Personal Information Full Name: Alfonso Sorrentino. Citizenship: Italian. Researcher unique identifier (ORCID): 0000-0002-5680-2999. Contact Information: Address: Dipartimento di Matematica, Universit`adegli Studi di Roma \Tor Vergata" Via della Ricerca Scientifica 1, 00133 Rome (Italy). Phone: (+39) 06 72594663 Email: [email protected] Website: http://www.mat.uniroma2.it/∼sorrenti • Research Interests Hamiltonian and Lagrangian systems: Aubry-Mather-Ma~n´etheory, KAM theory, weak KAM theory, Integrable systems, geodesic flows, Stability and Instability. Twist maps and symplectic maps: low-dimensional (topological) dynamics, Aubry-Mather theory. Billiards: dynamics, integrability, spectral properties, rigidity phenomena. Dissipative systems: conformally symplectic Aubry-Mather theory. Hamilton-Jacobi equation: Homogenization, Symplectic Homogenization, Hamilton-Jacobi on net- works and ramified spaces. Symplectic and contact geometry/topology: general theory, Hofer and Viterbo geometries, applica- tions to dynamics. • Education 2004 - 2008: Ph.D. in Mathematics, Princeton University (USA). Thesis Title: On the structure of action-minimizing sets for Lagrangian systems. Advisor: Prof. John N. Mather. Degree Committee: John N. Mather (President), Elon Lindenstrauss,Yakov Sinai and Bo0az Klartag. 2003 - 2004: M.A. in Mathematics, Princeton University (USA). Exam Committee: John Mather (President), Alice Chang and J´anosKoll´ar. 1998 - 2003: Laurea degree in Mathematics, Universit`adegli Studi \Roma Tre". Thesis Title: On smooth quasi-periodic solutions of Hamiltonian Systems. Supervisor: Prof. Luigi Chierchia. Evaluation: 110/110 cum laude. • Academic Positions 2014 - present: Associate Professor in Mathematical Analysis (01/A3, MAT/05) (tenured position) at Dipartimento di Matematica, Universit`adegli Studi di Roma \Tor Vergata", Rome (Italy). 2012 - 2014: Researcher in Mathematical Analysis MAT/05 (tenured position) at Dipartimento di Matematica e Fisica, Universit`adegli Studi \Roma Tre", Rome (Italy).
    [Show full text]
  • Ergodic Theory Plays a Key Role in Multiple Fields Steven Ashley Science Writer
    CORE CONCEPTS Core Concept: Ergodic theory plays a key role in multiple fields Steven Ashley Science Writer Statistical mechanics is a powerful set of professor Tom Ward, reached a key milestone mathematical tools that uses probability the- in the early 1930s when American mathema- ory to bridge the enormous gap between the tician George D. Birkhoff and Austrian-Hun- unknowable behaviors of individual atoms garian (and later, American) mathematician and molecules and those of large aggregate sys- and physicist John von Neumann separately tems of them—a volume of gas, for example. reconsidered and reformulated Boltzmann’ser- Fundamental to statistical mechanics is godic hypothesis, leading to the pointwise and ergodic theory, which offers a mathematical mean ergodic theories, respectively (see ref. 1). means to study the long-term average behavior These results consider a dynamical sys- of complex systems, such as the behavior of tem—whetheranidealgasorothercomplex molecules in a gas or the interactions of vi- systems—in which some transformation func- brating atoms in a crystal. The landmark con- tion maps the phase state of the system into cepts and methods of ergodic theory continue its state one unit of time later. “Given a mea- to play an important role in statistical mechan- sure-preserving system, a probability space ics, physics, mathematics, and other fields. that is acted on by the transformation in Ergodicity was first introduced by the a way that models physical conservation laws, Austrian physicist Ludwig Boltzmann laid Austrian physicist Ludwig Boltzmann in the what properties might it have?” asks Ward, 1870s, following on the originator of statisti- who is managing editor of the journal Ergodic the foundation for modern-day ergodic the- cal mechanics, physicist James Clark Max- Theory and Dynamical Systems.Themeasure ory.
    [Show full text]
  • Fermat's Last Theorem, a Theorem at Last
    August 1993 MAA FOCUS Fermat’s Last Theorem, that one could understand the elliptic curve given by the equation a Theorem at Last 2 n n y = x(x − a )( x + b ) Keith Devlin, Fernando Gouvêa, and Andrew Granville in the way proposed by Taniyama. After defying all attempts at a solution for Wiles’ approach comes from a somewhat Following an appropriate re-formulation 350 years, Fermat’s Last Theorem finally different direction, and rests on an amazing by Jean-Pierre Serre in Paris, Kenneth took its place among the known theorems of connection, established during the last Ribet in Berkeley strengthened Frey’s mathematics in June of this year. decade, between the Last Theorem and the original concept to the point where it was theory of elliptic curves, that is, curves possible to prove that the existence of a On June 23, during the third of a series of determined by equations of the form counter example to the Last Theorem 2 3 lectures at a conference held at the Newton y = x + ax + b, would lead to the existence of an elliptic Institute in Cambridge, British curve which could not be modular, and mathematician Dr. Andrew Wiles, of where a and b are integers. hence would contradict the Shimura- Princeton University, sketched a proof of the Taniyama-Weil conjecture. Shimura-Taniyama-Weil conjecture for The path that led to the June 23 semi-stable elliptic curves. As Kenneth announcement began in 1955 when the This is the point where Wiles entered the Ribet, of the University of California at Japanese mathematician Yutaka Taniyama picture.
    [Show full text]
  • Associating Abelian Varieties to Weight-2 Modular Forms: the Eichler-Shimura Construction
    Ecole´ Polytechnique Fed´ erale´ de Lausanne Master’s Thesis in Mathematics Associating abelian varieties to weight-2 modular forms: the Eichler-Shimura construction Author: Corentin Perret-Gentil Supervisors: Prof. Akshay Venkatesh Stanford University Prof. Philippe Michel EPF Lausanne Spring 2014 Abstract This document is the final report for the author’s Master’s project, whose goal was to study the Eichler-Shimura construction associating abelian va- rieties to weight-2 modular forms for Γ0(N). The starting points and main resources were the survey article by Fred Diamond and John Im [DI95], the book by Goro Shimura [Shi71], and the book by Fred Diamond and Jerry Shurman [DS06]. The latter is a very good first reference about this sub- ject, but interesting points are sometimes eluded. In particular, although most statements are given in the general setting, the book mainly deals with the particular case of elliptic curves (i.e. with forms having rational Fourier coefficients), with little details about abelian varieties. On the other hand, Chapter 7 of Shimura’s book is difficult, according to the author himself, and the article by Diamond and Im skims rapidly through the subject, be- ing a survey. The goal of this document is therefore to give an account of the theory with intermediate difficulty, accessible to someone having read a first text on modular forms – such as [Zag08] – and with basic knowledge in the theory of compact Riemann surfaces (see e.g. [Mir95]) and algebraic geometry (see e.g. [Har77]). This report begins with an account of the theory of abelian varieties needed for what follows.
    [Show full text]