2014 Fields Medals
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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. -
Amie Wilkinson 1
Amie Wilkinson 1 Amie Wilkinson Department of Mathematics, University of Chicago 5734 S. University Avenue Chicago, Illinois 60637 (773) 702-7337 [email protected] ACADEMIC POSITIONS • Professor of Mathematics, University of Chicago, 2012 { present. • Professor of Mathematics, Northwestern University, 2005 { 2012. • Associate Professor of Mathematics, Northwestern University, 2002 { 2005. • Assistant Professor of Mathematics, Northwestern University, 1999 { 2002. • Boas Assistant Professor of Mathematics, Northwestern University, 1996 { 1999. • Benjamin Peirce Instructor, Harvard University, 1995 { 1996. Visiting Positions • Visiting Professor, University of Chicago, Fall and Spring Quarters 2003{2004, Fall Quarter 2011. • Professeur Invit´e, Universit´ede Bourgogne, May 2002 and Sept 2003. • Member, Institut des Hautes Etudes Scientifiques (IHES), Summer 1993, 1996, 1998. • Visitor, IBM T.J. Watson Labs, Yorktown NY, Winter 1992 and 1994, Summer 1997, 1998, 2000, and 2001. • Graduate Research Assistant, The Center for Nonlinear Studies, Los Alamos National Laboratories, Summer, 1992. EDUCATION • Ph.D. in Mathematics, University of California, Berkeley, May 1995 • A.B. in Mathematics, Harvard University, June 1989 DATE OF BIRTH April, 1968. U.S. citizen. Amie Wilkinson 2 RESEARCH INTERESTS • Ergodic theory and smooth dynamical systems • Geometry and regularity of foliations • Actions of discrete groups on manifolds • Dynamical systems of geometric origin GRANTS, FELLOWSHIPS AND AWARDS • Levi L. Conant Prize, 2020. • Foreign Member, Academia Europaea, 2019. • Fellow of the American Mathematical Society, 2013. • Ruth Lyttle Satter Prize, 2011. • NSF Grant \Ergodicity, Rigidity, and the Interplay Between Chaotic and Regular Dynamics" $758,242, 2018{2021. • NSF Grant \Innovations in Bright Beam Science" (co-PI) $680,000, 2015{2018. • NSF Grant \RTG: Geometry and topology at the University of Chicago" (co-PI) $1,377,340, 2014{2019. -
NEWSLETTER No
NEWSLETTER No. 455 February 2016 LMS INAUGURAL HIRST LECTURE: PROFESSOR EDMUND F. ROBERTSON he Society is pleased to announce that MacTutor History of Mathematics website has TProfessor Edmund F. Robertson (St Andrews) now become an important resource for those in- will give the inaugural Hirst Lecture at St terested in the history of mathematics. It contains Andrews on Wednesday 20 April 2016. Mark over 2,800 biographies of mathematicians and is McCartney (University of Ulster) will give an ac- used across the world by schoolchildren, under- companying lecture. graduates, graduates and their teachers. The Hirst Lecture celebrates the joint award of The Hirst Prize and Lectureship are named after the Hirst Prize & Lectureship, in the 150th Anni- Thomas A. Hirst, 5th President of the London versary year of the London Mathematical Society, Mathematical Society from 1872-1874. The prize to Professor Edmund Robertson (St Andrews) and is awarded in recognition of original and inno- Dr John O’Connor (St Andrews) for their creation, vative work in the history of mathematics, which development and maintenance of the MacTutor may be in any medium. History of Mathematics website (www-history. In 2015, the Council of the Society agreed to mcs.st-and.ac.uk). continue the Hirst Prize and Lectureship on a Originally developed in the early 1990s to enrich biennial basis with the next award to be made the Mathematical MacTutor System that supports in 2018 and the lecture to be given at a Society teaching mathematics to undergraduates, the Meeting in 2019. -
CURRENT EVENTS BULLETIN Friday, January 8, 2016, 1:00 PM to 5:00 PM Room 4C-3 Washington State Convention Center Joint Mathematics Meetings, Seattle, WA
A MERICAN M ATHEMATICAL S OCIETY CURRENT EVENTS BULLETIN Friday, January 8, 2016, 1:00 PM to 5:00 PM Room 4C-3 Washington State Convention Center Joint Mathematics Meetings, Seattle, WA 1:00 PM Carina Curto, Pennsylvania State University What can topology tell us about the neural code? Surprising new applications of what used to be thought of as “pure” mathematics. 2:00 PM Yuval Peres, Microsoft Research and University of California, Berkeley, and Lionel Levine, Cornell University Laplacian growth, sandpiles and scaling limits Striking large-scale structure arising from simple cellular automata. 3:00 PM Timothy Gowers, Cambridge University Probabilistic combinatorics and the recent work of Peter Keevash The major existence conjecture for combinatorial designs has been proven! 4:00 PM Amie Wilkinson, University of Chicago What are Lyapunov exponents, and why are they interesting? A basic tool in understanding the predictability of physical systems, explained. Organized by David Eisenbud, Mathematical Sciences Research Institute Introduction to the Current Events Bulletin Will the Riemann Hypothesis be proved this week? What is the Geometric Langlands Conjecture about? How could you best exploit a stream of data flowing by too fast to capture? I think we mathematicians are provoked to ask such questions by our sense that underneath the vastness of mathematics is a fundamental unity allowing us to look into many different corners -- though we couldn't possibly work in all of them. I love the idea of having an expert explain such things to me in a brief, accessible way. And I, like most of us, love common-room gossip. -
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. -
President's Report
Newsletter Volume 43, No. 3 • mAY–JuNe 2013 PRESIDENT’S REPORT Greetings, once again, from 35,000 feet, returning home from a major AWM conference in Santa Clara, California. Many of you will recall the AWM 40th Anniversary conference held in 2011 at Brown University. The enthusiasm generat- The purpose of the Association ed by that conference gave rise to a plan to hold a series of biennial AWM Research for Women in Mathematics is Symposia around the country. The first of these, the AWM Research Symposium 2013, took place this weekend on the beautiful Santa Clara University campus. • to encourage women and girls to study and to have active careers The symposium attracted close to 150 participants. The program included 3 plenary in the mathematical sciences, and talks, 10 special sessions on a wide variety of topics, a contributed paper session, • to promote equal opportunity and poster sessions, a panel, and a banquet. The Santa Clara campus was in full bloom the equal treatment of women and and the weather was spectacular. Thankfully, the poster sessions and coffee breaks girls in the mathematical sciences. were held outside in a courtyard or those of us from more frigid climates might have been tempted to play hooky! The event opened with a plenary talk by Maryam Mirzakhani. Mirzakhani is a professor at Stanford and the recipient of multiple awards including the 2013 Ruth Lyttle Satter Prize. Her talk was entitled “On Random Hyperbolic Manifolds of Large Genus.” She began by describing how to associate a hyperbolic surface to a graph, then proceeded with a fascinating discussion of the metric properties of surfaces associated to random graphs. -
What Are Lyapunov Exponents, and Why Are They Interesting?
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 1, January 2017, Pages 79–105 http://dx.doi.org/10.1090/bull/1552 Article electronically published on September 6, 2016 WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING? AMIE WILKINSON Introduction At the 2014 International Congress of Mathematicians in Seoul, South Korea, Franco-Brazilian mathematician Artur Avila was awarded the Fields Medal for “his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.”1 Although it is not explicitly mentioned in this citation, there is a second unify- ing concept in Avila’s work that is closely tied with renormalization: Lyapunov (or characteristic) exponents. Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odinger operators. Here we take the op- portunity to explore these areas and reveal some underlying themes connecting exponents, chaotic dynamics and renormalization. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats: the iterated barycentric subdivision of a triangle. When the midpoint of each side of a triangle is connected to its opposite vertex by a line segment, the three resulting segments meet in a point in the interior of the triangle. The barycentric subdivision of a triangle is the collection of 6 smaller triangles determined by these segments and the edges of the original triangle: Figure 1. Barycentric subdivision. Received by the editors August 2, 2016. -
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. -
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. -
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. -
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.