Mathematical Olympiad & Other Scholarship, Research Programmes
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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. -
Arithmetic Equivalence and Isospectrality
ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY ANDREW V.SUTHERLAND ABSTRACT. In these lecture notes we give an introduction to the theory of arithmetic equivalence, a notion originally introduced in a number theoretic setting to refer to number fields with the same zeta function. Gassmann established a direct relationship between arithmetic equivalence and a purely group theoretic notion of equivalence that has since been exploited in several other areas of mathematics, most notably in the spectral theory of Riemannian manifolds by Sunada. We will explicate these results and discuss some applications and generalizations. 1. AN INTRODUCTION TO ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY Let K be a number field (a finite extension of Q), and let OK be its ring of integers (the integral closure of Z in K). The Dedekind zeta function of K is defined by the Dirichlet series X s Y s 1 ζK (s) := N(I)− = (1 N(p)− )− I OK p − ⊆ where the sum ranges over nonzero OK -ideals, the product ranges over nonzero prime ideals, and N(I) := [OK : I] is the absolute norm. For K = Q the Dedekind zeta function ζQ(s) is simply the : P s Riemann zeta function ζ(s) = n 1 n− . As with the Riemann zeta function, the Dirichlet series (and corresponding Euler product) defining≥ the Dedekind zeta function converges absolutely and uniformly to a nonzero holomorphic function on Re(s) > 1, and ζK (s) extends to a meromorphic function on C and satisfies a functional equation, as shown by Hecke [25]. The Dedekind zeta function encodes many features of the number field K: it has a simple pole at s = 1 whose residue is intimately related to several invariants of K, including its class number, and as with the Riemann zeta function, the zeros of ζK (s) are intimately related to the distribution of prime ideals in OK . -
Tarjan Transcript Final with Timestamps
A.M. Turing Award Oral History Interview with Robert (Bob) Endre Tarjan by Roy Levin San Mateo, California July 12, 2017 Levin: My name is Roy Levin. Today is July 12th, 2017, and I’m in San Mateo, California at the home of Robert Tarjan, where I’ll be interviewing him for the ACM Turing Award Winners project. Good afternoon, Bob, and thanks for spending the time to talk to me today. Tarjan: You’re welcome. Levin: I’d like to start by talking about your early technical interests and where they came from. When do you first recall being interested in what we might call technical things? Tarjan: Well, the first thing I would say in that direction is my mom took me to the public library in Pomona, where I grew up, which opened up a huge world to me. I started reading science fiction books and stories. Originally, I wanted to be the first person on Mars, that was what I was thinking, and I got interested in astronomy, started reading a lot of science stuff. I got to junior high school and I had an amazing math teacher. His name was Mr. Wall. I had him two years, in the eighth and ninth grade. He was teaching the New Math to us before there was such a thing as “New Math.” He taught us Peano’s axioms and things like that. It was a wonderful thing for a kid like me who was really excited about science and mathematics and so on. The other thing that happened was I discovered Scientific American in the public library and started reading Martin Gardner’s columns on mathematical games and was completely fascinated. -
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. -
June 2014 Society Meetings Society and Events SHEPHARD PRIZE: NEW PRIZE Meetings for MATHEMATICS 2014 and Events Following a Very Generous Tions Open in Late 2014
LONDONLONDON MATHEMATICALMATHEMATICAL SOCIETYSOCIETY NEWSLETTER No. 437 June 2014 Society Meetings Society and Events SHEPHARD PRIZE: NEW PRIZE Meetings FOR MATHEMATICS 2014 and Events Following a very generous tions open in late 2014. The prize Monday 16 June donation made by Professor may be awarded to either a single Midlands Regional Meeting, Loughborough Geoffrey Shephard, the London winner or jointly to collaborators. page 11 Mathematical Society will, in 2015, The mathematical contribution Friday 4 July introduce a new prize. The prize, to which an award will be made Graduate Student to be known as the Shephard must be published, though there Meeting, Prize will be awarded bienni- is no requirement that the pub- London ally. The award will be made to lication be in an LMS-published page 8 a mathematician (or mathemati- journal. Friday 4 July cians) based in the UK in recog- Professor Shephard himself is 1 Society Meeting nition of a specific contribution Professor of Mathematics at the Hardy Lecture to mathematics with a strong University of East Anglia whose London intuitive component which can be main fields of interest are in page 9 explained to those with little or convex geometry and tessella- Wednesday 9 July no knowledge of university math- tions. Professor Shephard is one LMS Popular Lectures ematics, though the work itself of the longest-standing members London may involve more advanced ideas. of the LMS, having given more page 17 The Society now actively en- than sixty years of membership. Tuesday 19 August courages members to consider The Society wishes to place on LMS Meeting and Reception nominees who could be put record its thanks for his support ICM 2014, Seoul forward for the award of a in the establishment of the new page 11 Shephard Prize when nomina- prize. -
Press Release
Press release 13 August 2014 Two Fields Medals 2014 awarded to ERC laureates The 2014 Fields Medals were awarded today to four outstanding mathematicians, of whom two are grantees of the European Research Council (ERC): Prof. Artur Avila (Brazil-France), an ERC Starting grant holder since 2010, and Prof. Martin Hairer (Austria) has been selected for funding under an ERC Consolidator grant in 2013. They received the prize respectively for their work on dynamical systems and probability, and on stochastic analysis. The other two laureates are Prof. Manjul Barghava (Canada-US) and Prof. Maryam Mirzakhani (Iran). The Medals were announced at the International Congress of Mathematicians (ICM) taking place from 13 – 21 August in Seoul, South Korea. On the occasion of the announcement, ERC President Prof. Jean-Pierre Bourguignon, a mathematician himself, said: “On behalf of the ERC Scientific Council, I would like to congratulate warmly all four Fields Medallists for their outstanding contributions to the field of mathematics. My special congratulations go to Artur Avila and Martin Hairer, who are both brilliant scientists supported by the ERC. We are happy to see that their remarkable talent in the endless frontiers of science has been recognised. The Fields Medals awarded today are also a sign that the ERC continues to identity and fund the most promising researchers across Europe; this is true not only in mathematics but in all scientific disciplines.” EU Commissioner for Research, Innovation and Science, Máire Geoghegan-Quinn, said: “I would like to congratulate the four laureates of the Fields Medal announced today. The Fields Medal, the highest international distinction for young mathematicians, is a well- deserved honour for hardworking and creative young researchers who push the boundaries of knowledge. -
Higher Algebraic K-Theory I
1 Higher algebraic ~theory: I , * ,; Daniel Quillen , ;,'. ··The·purpose of..thispaper.. is.to..... develop.a.higher. X..,theory. fpJ;' EiddUiy!!. categQtl~ ... __ with euct sequences which extends the ell:isting theory of ths Grothsndieck group in a natural wll7. To describe' the approach taken here, let 10\ be an additive category = embedded as a full SUbcategory of an abelian category A, and assume M is closed under , = = extensions in A. Then one can form a new category Q(M) having the same objects as ')0\ , = =, = but :in which a morphism from 101 ' to 10\ is taken to be an isomorphism of MI with a subquotient M,IM of M, where MoC 101, are aubobjects of M such that 101 and MlM, o 0 are objects of ~. Assuming 'the isomorphism classes of objects of ~ form a set, the, cstegory Q(M)= has a classifying space llQ(M)= determined up to homotopy equivalence. One can show that the fundamental group of this classifying spacs is canonically isomor- phic to the Grothendieck group of ~ which motivates dsfining a ssquenoe of X-groups by the formula It is ths goal of the present paper to show that this definition leads to an interesting theory. The first part pf the paper is concerned with the general theory of these X-groups. Section 1 contains various tools for working .~th the classifying specs of a small category. It concludes ~~th an important result which identifies ·the homotopy-theoretic fibre of the map of classifying spaces induced by a.functor. In X-theory this is used to obtain long exsct sequences of X-groups from the exact homotopy sequence of a map. -
ICTS POSTER Outside Bangalore
T A T A I N S T I T U T E O F F U N D A M E N T A L R E S E A R C H A HOMI BHABHA BIRTH CENTENARY & ICTS INAUGURAL EVENT International Centre Theoretical Sciences science without bo28 Decemberun 2009d29 -a 31 Decemberri e2009s Satish Dhawan Auditorium Faculty Hall Indian Institute of Science, Bangalore. www.icts.res.in/program/icts-ie INVITED SPEAKERS / PANELISTS INCLUDE FOUNDATION STONE CEREMONY Siva Athreya ISI, Bangalore OF ICTS CAMPUS Naama Barkai Weizmann Institute The foundation stone will be unveiled by Manjul Bhargava Princeton University Prof. C N R Rao, FRS 4:00 pm, 28 December 2009 Édouard Brézin École Normale Supérieure Amol Dighe TIFR Michael Green DAMTP, Cambridge Chandrashekhar Khare UCLA Yamuna Krishnan NCBS-TIFR Lyman Page Princeton University Jaikumar Radhakrishnan TIFR C. S. Rajan TIFR Sriram Ramaswamy IISc G. Rangarajan IISc C. N. R. Rao JNCASR Subir Sachdev Harvard University K. Sandeep CAM-TIFR Sriram Shastry UC Santa Cruz PUBLIC LECTURES Ashoke Sen HRI J. N. Tata Auditorium, IISc (FREE AND OPEN TO ALL) Anirvan Sengupta Rutgers University K. R. Sreenivasan Abdus Salam ICTP Michael Atiyah University of Edinburgh Andrew Strominger Harvard University Truth and Beauty in Mathematics and Physics 5:30 pm, 27 December 2009 Raman Sundrum Johns Hopkins University Ajay Sood IISc David Gross KITP, Santa Barbara The Role of Theory in Science Tarun Souradeep IUCAA 5:30 pm, 28 December 2009 Eitan Tadmor University of Maryland Albert Libchaber Rockefeller University Sandip Trivedi TIFR The Origin of Life: from Geophysics to Biology? Mukund Thattai NCBS-TIFR 5:30 pm, 30 December 2009 S. -
Roth's Theorem in the Primes
Annals of Mathematics, 161 (2005), 1609–1636 Roth’s theorem in the primes By Ben Green* Abstract We show that any set containing a positive proportion of the primes con- tains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We de- rive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1. Introduction Arguably the second most famous result of Klaus Roth is his 1953 upper bound [21] on r3(N), defined 17 years previously by Erd˝os and Tur´an to be the cardinality of the largest set A [N] containing no nontrivial 3-term arithmetic ⊆ progression (3AP). Roth was the first person to show that r3(N) = o(N). In fact, he proved the following quantitative version of this statement. Proposition 1.1 (Roth). r (N) N/ log log N. 3 " There was no improvement on this bound for nearly 40 years, until Heath- Brown [15] and Szemer´edi [22] proved that r N(log N) c for some small 3 " − positive constant c. Recently Bourgain [6] provided the best bound currently known. Proposition 1.2 (Bourgain). r (N) N (log log N/ log N)1/2. 3 " *The author is supported by a Fellowship of Trinity College, and for some of the pe- riod during which this work was carried out enjoyed the hospitality of Microsoft Research, Redmond WA and the Alfr´ed R´enyi Institute of the Hungarian Academy of Sciences, Bu- dapest. -
All That Math Portraits of Mathematicians As Young Researchers
Downloaded from orbit.dtu.dk on: Oct 06, 2021 All that Math Portraits of mathematicians as young researchers Hansen, Vagn Lundsgaard Published in: EMS Newsletter Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hansen, V. L. (2012). All that Math: Portraits of mathematicians as young researchers. EMS Newsletter, (85), 61-62. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. -
Klaus Friedrich Roth, 1925–2015 the Godfrey Argent Studio C
e Bull. London Math. Soc. 50 (2018) 529–560 C 2017 Authors doi:10.1112/blms.12143 OBITUARY Klaus Friedrich Roth, 1925–2015 The Godfrey Argent Studio C Klaus Friedrich Roth, who died in Inverness on 10 November 2015 aged 90, made fundamental contributions to different areas of number theory, including diophantine approximation, the large sieve, irregularities of distribution and what is nowadays known as arithmetic combinatorics. He was the first British winner of the Fields Medal, awarded in 1958 for his solution in 1955 of the famous Siegel conjecture concerning approximation of algebraic numbers by rationals. He was elected a member of the London Mathematical Society on 17 May 1951, and received its De Morgan Medal in 1983. 1. Life and career Klaus Roth, son of Franz and Matilde (n´ee Liebrecht), was born on 29 October 1925, in the German city of Breslau, in Lower Silesia, Prussia, now Wroclaw in Poland. To escape from Nazism, he and his parents moved to England in 1933 and settled in London. He would recall that the flight from Berlin to London took eight hours and landed in Croydon. Franz, a solicitor by training, had suffered from gas poisoning during the First World War, and died a few years after their arrival in England. Roth studied at St Paul’s School between 1937 and 1943, during which time the school was relocated to Easthampstead Park, near Crowthorne in Berkshire, as part of the wartime evac- uation of London. There he excelled in mathematics and chess, and one master, Mr Dowswell, observed interestingly that he possessed complete intellectual honesty. -
Program of the Sessions Washington, District of Columbia, January 5–8, 2009
Program of the Sessions Washington, District of Columbia, January 5–8, 2009 MAA Short Course on Data Mining and New Saturday, January 3 Trends in Teaching Statistics (Part I) 8:00 AM –4:00PM Delaware Suite A, Lobby Level, Marriott Organizer: Richard D. De Veaux, Williams College AMS Short Course on Quantum Computation 8:00AM Registration and Quantum Information (Part I) 9:00AM Math is music—statistics is literature. (5) What are the challenges of teaching 8:00 AM –5:00PM Virginia Suite A, statistics, and why is it different from Lobby Level, Marriott mathematics? Richard D. De Veaux, Williams College Organizer: Samuel J. Lomonaco, 10:30AM Break. University of Maryland Baltimore County 10:45AM What does the introductory course look (6) like in 2009? How technology has 8:00AM Registration. changed what we do in introductory 9:00AM A Rosetta Stone for quantum computing. statistics for the non-math/science (1) Samuel Lomonaco,Universityof student. Maryland Baltimore County Richard D. De Veaux, Williams College 10:15AM Break. 1:00PM What does the math-based introductory (7) course look like in 2009? How do 10:45AM Quantum algorithms. we merge mathematical concepts (2) Peter Shor, Massachusetts Institute of into the introductory course for the Technology math/science student? How does 2:00PM Concentration of measure effects in statistical programming fit in? (3) quantum information. Richard D. De Veaux, Williams College Patrick Hayden, McGill University 2:30PM Break. 3:15PM Break. 2:45PM Introduction to Modeling. Regression and 3:45PM Quantum error correction and fault (8) ANOVA. Overview: How much to teach (4) tolerance.