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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. -
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. -
Read Press Release
The Work of Artur Avila Artur Avila has made outstanding contributions to dynamical systems, analysis, and other areas, in many cases proving decisive results that solved long-standing open problems. A native of Brazil who spends part of his time there and part in France, he combines the strong mathematical cultures and traditions of both countries. Nearly all his work has been done through collaborations with some 30 mathematicians around the world. To these collaborations Avila brings formidable technical power, the ingenuity and tenacity of a master problem-solver, and an unerring sense for deep and significant questions. Avila's achievements are many and span a broad range of topics; here we focus on only a few highlights. One of his early significant results closes a chapter on a long story that started in the 1970s. At that time, physicists, most notably Mitchell Feigenbaum, began trying to understand how chaos can arise out of very simple systems. Some of the systems they looked at were based on iterating a mathematical rule such as 3x(1−x). Starting with a given point, one can watch the trajectory of the point under repeated applications of the rule; one can think of the rule as moving the starting point around over time. For some maps, the trajectories eventually settle into stable orbits, while for other maps the trajectories become chaotic. Out of the drive to understand such phenomena grew the subject of discrete dynamical systems, to which scores of mathematicians contributed in the ensuing decades. Among the central aims was to develop ways to predict long-time behavior. -
Arithmetic Sums of Nearly Affine Cantor Sets
UNIVERSITY OF CALIFORNIA, IRVINE Arithmetic Sums of Nearly Affine Cantor Sets DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Mathematics by Scott Northrup Dissertation Committee: Associate Professor Anton Gorodetski, Chair Professor Svetlana Jitomirskaya Associate Professor Germ´anA. Enciso 2015 c 2015 Scott Northrup DEDICATION To Cynthia and Max. ii TABLE OF CONTENTS Page LIST OF FIGURES iv ACKNOWLEDGMENTS v CURRICULUM VITAE vi ABSTRACT OF THE DISSERTATION vii Introduction 1 1 Outline 7 1.1 Statement of the Results . 7 1.2 Organization of the Thesis . 9 2 Cantor Sets 11 2.1 Dynamically Defined Cantor Sets . 11 2.2 Dimension Theory . 15 2.3 The Palis Conjecture . 18 2.4 Middle-α Cantor Sets . 19 2.5 Homogeneous Self-Similar Cantor Sets . 21 2.6 A Positive Answer to the Palis Conjecture . 22 2.7 The dimH C1 + dimH C2 < 1 Case . 23 3 Results for Nearly Affine Cantor Sets 26 3.1 Absolute Continuity of Convolutions . 26 3.2 Theorem 1.1 . 27 3.3 Construction of Measures for K and Cλ ..................... 28 3.4 Verifying the Conditions of Proposition 3.1 . 32 3.5 Corollaries . 41 4 Open Questions 42 Bibliography 44 A Proof of Proposition 3.1 51 iii LIST OF FIGURES Page 1 The horseshoe map; ' contracts the square horizontally and expands it verti- cally, before folding it back onto itself. The set of points invariant under this map will form a hyperbolic set, called Smale's Horseshoe............ 4 2.1 The Horseshoe map, again . 12 2.2 A few iterations of the Stable Manifold . -
Arxiv:1609.08664V2 [Math-Ph] 9 Dec 2017 Yhftde N[ in Hofstadter by Ae[ Case Atrpern Sntitne O Publication
UNIVERSAL HIERARCHICAL STRUCTURE OF QUASIPERIODIC EIGENFUNCTIONS SVETLANA JITOMIRSKAYA AND WENCAI LIU Abstract. We determine exact exponential asymptotics of eigenfunctions and of corre- sponding transfer matrices of the almost Mathieu operators for all frequencies in the lo- calization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics liter- ature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the cor- responding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied. 1. Introduction A very captivating question and a longstanding theoretical challenge in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describ- ing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. Such structure was first predicted in the work of Azbel in 1964 [11]. It was numerically confirmed through the famous butterfly plot and further argued for by Hofstadter in [29], for the spectrum of the almost Mathieu operator. This was even before this model was linked to the integer quantum Hall effect [48] and other important phenom- ena. Mathematically, it is known that the spectrum is a Cantor set for all irrational fluxes [5], and moreover, even all gaps predicted by the gap labeling are open in the non-critical case [6, 8]. -
Lecture Notes in Physics
Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg, Germany G. Hasinger, Garching, Germany K. Hepp, Zürich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Zürich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Lipowsky, Golm, Germany H. v. Löhneysen, Karlsruhe, Germany I. Ojima, Kyoto, Japan D. Sornette, Nice, France, and Zürich, Switzerland S. Theisen, Golm, Germany W. Weise, Garching, Germany J. Wess, München, Germany J. Zittartz, Köln, Germany The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching – quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced grad- uate textbooks and the forefront of research to serve the following purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic; • to serve as an accessible introduction to the field to postgraduate students and nonspe- cialist researchers from related areas; • to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Pro- ceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive is available at springerlink.com. -
Yuki Takahashi
Yuki Takahashi Contact Saitama University [email protected] Information Department of Mathematics http://www.math.uci.edu/~takahasy/ Research Dynamical Systems, Spectral Theory and Fractal Geometry. Interests Employment Department of Mathematics, Saitama University • Assistant Professor, October 2020 - present Department of Mathematics, Michigan State University • Visiting Assistant Professor, January 2020 - August 2020 (research mentor: I. Kachkovskiy) AIMR Mathematical Science Group, Tohoku University • Assistant Professor (non tenure-track), April 2019 - December 2019 (Chiba research group) Department of Mathematics, Bar-Ilan University • Postdoc, June 2017 - March 2019 (supervisor: B. Solomyak) Education Department of Mathematics, University of California, Irvine • Ph.D. in Mathematics, May 2017 (advisor: A. Gorodetski) Ph.D. Thesis: Sums and products of Cantor sets and separable two-dimensional quasicrystal models The University of Tokyo • M.S. in Mathematics, March 2012 Master's Thesis: Irregular solutions of the periodic discrete Toda lattice equation (in Japanese) • B.A. in Mathematics, March 2009 Grants Spring 2020 Leibniz Fellowship Grant (Mathematisches Forschungsinstitut Oberwolfach) Fall 2019 FY2019 WPI-AIMR Fusion Research (Microfluidic, energy, and spin devices with fractal space-filling geometries, together with H. Izuchi, H. Kai and K. Kizu) Winter 2018 Leibniz Fellowship Grant (declined) (Mathematisches Forschungsinstitut Oberwolfach) Fall 2017 Postdoctoral Fellowship Grant (Institut Mittag-Leffler, postdoctoral fellowship) Honors and Spring 2017 Dissertation Fellowship (UC Irvine, department fellowship) Awards Winter 2017 Department Fellowship (UC Irvine, department fellowship) Spring 2014 Connelly Award (UC Irvine, department award) Spring 2013 Euler Outstanding Promise as a Graduate Student Award (UC Irvine, department award) Papers (in 1. Y. Takahashi, On the invariant measures for Iterated Function Systems with in- preparation) verses. -
Joint Summer Research Conferences in the Mathematical Sciences
jsrc-sri.qxp 9/24/98 8:56 AM Page 1435 Conferences All requests will be forwarded to the appropriate orga- Joint Summer nizing committee for consideration. In late April all ap- plicants will receive formal invitations (including specific Research Conferences offers of support if applicable), a brochure of conference information, program information known to date, along with information on travel and dormitories and other local in the Mathematical housing. All participants will be required to pay a nomi- nal conference fee. Sciences Questions concerning the scientific program should be University of Colorado addressed to the organizers. Questions of a nonscientific Boulder, Colorado nature should be directed to the Summer Research Con- June 13–July 1, 1999 ferences coordinator at the address provided above. Please watch http://www.ams.org/meetings/ for future de- The 1999 Joint Summer Research Conferences will be held velopments about these conferences. at the University of Colorado, Boulder, Colorado, from Lectures begin on Sunday morning and run through June 13–July 1, 1999. The topics and organizers for the Thursday. Check-in for housing begins on Saturday. No seven conferences were selected by a committee repre- lectures are held on Saturday. senting the AMS, the Institute of Mathematical Sciences (IMS), and the Society for Industrial and Applied Math- ematics (SIAM), whose members at the time were Alejan- From Manifolds to Singular Varieties dro Adem, David Brydges, Percy Deift, James W. Demmel, Sunday, June 13–Thursday, June 17, 1999 Dipak Dey, Tom Diciccio, Steven Hurder, Alan F. Karr, Bar- bara Keyfitz, W. Brent Lindquist, Andre Manitius, and Bart Sylvain E. -
2005 Satter Prize, Volume 52, Number 4
2005 Satter Prize The 2005 Ruth Lyttle Satter Prize in Mathematics sketch, and the awardee’s re- was awarded at the 111th Annual Meeting of the sponse upon receiving the prize. AMS in Atlanta in January 2005. The Satter Prize is awarded every two years to Citation recognize an outstanding contribution to mathe- The Ruth Lyttle Satter Prize in Math- matics research by a woman in the previous five ematics is awarded to Svetlana Jit- years. Established in 1990 with funds donated by omirskaya for her pioneering work Joan S. Birman, the prize honors the memory of Bir- on non-perturbative quasiperiodic man’s sister, Ruth Lyttle Satter. Satter earned a localization, in particular for re- bachelor’s degree in mathematics and then joined sults in her papers (1) “Metal-in- the research staff at AT&T Bell Laboratories dur- sulator transition for the almost ing World War II. After raising a family, she received Mathieu operator”, Ann. of Math. a Ph.D. in botany at the age of forty-three from the (2) 150 (1999), no. 3, 1159–1175, Svetlana Jitomirskaya University of Connecticut at Storrs, where she later and (2) with J. Bourgain, “Ab- became a faculty member. Her research on the bi- solutely continuous spectrum for 1D quasiperiodic ological clocks in plants earned her recognition in operators”, Invent. Math. 148 (2002), no. 3, 453–463. the U.S. and abroad. Birman requested that the In her Annals paper, she developed a non-perturba- prize be established to honor her sister’s commit- tive approach to quasiperiodic localization and solved ment to research and to encourage women in sci- the long-standing Aubry-Andre conjecture on the al- ence. -
Quasiperiodic Schrödinger Operators
Mathematisches Forschungsinstitut Oberwolfach Report No. 17/2012 DOI: 10.4171/OWR/2012/17 Arbeitsgemeinschaft: Quasiperiodic Schr¨odinger Operators Organised by Artur Avila, Rio de Janeiro/Paris David Damanik, Houston Svetlana Jitomirskaya, Irvine April 1st – April 7th, 2012 Abstract. This Arbeitsgemeinschaft discussed the spectral properties of quasi-periodic Schr¨odinger operators in one space dimension. After present- ing background material on Schr¨odinger operators with dynamically defined potentials and some results about certain classes of dynamical systems, the recently developed global theory of analytic one-frequency potentials was dis- cussed in detail. This was supplemented by presentations on an important special case, the almost Mathieu operator, and results showing phenomena exhibited outside the analytic category. Mathematics Subject Classification (2000): 47B36, 81Q10. Introduction by the Organisers This Arbeitsgemeinschaft was organized by Artur Avila (Paris 7 and IMPA Rio de Janeiro), David Damanik (Rice), and Svetlana Jitomirskaya (UC Irvine) and held April 1–7, 2012. There were 31 participants in total, of whom 26 gave talks. The objective was to discuss quasiperiodic Schr¨odinger operators of the form [Hψ](n)= ψ(n +1)+ ψ(n 1)+ V (n)ψ(n) − with potential V : Z R given by V (n) = f(ω + nα), where ω, α Tk = k k → ∈ R /Z , λ R and α = (α1,...,αk) is such that 1, α1,...,αk are independent over the rational∈ numbers, and f : Tk R is assumed to be at least continuous. H acts on the Hilbert space ℓ2(Z)→ as a bounded self-adjoint operator. The itH associated unitary group e− t R describes the evolution of a quantum particle subjected to the quasiperiodic{ environment} ∈ given by V . -
Vii. Communication of the Mathematical Sciences 135 Viii
¡ ¢£¡ ¡ ¤ ¤ ¥ ¦ ¡ The Pacific Institute for the Mathematical Sciences Our Mission Our Community The Pacific Institute for the Mathematical Sciences PIMS is a partnership between the following organi- (PIMS) was created in 1996 by the community zations and people: of mathematical scientists in Alberta and British Columbia and in 2000, they were joined in their en- The six participating universities (Simon Fraser deavour by their colleagues in the State of Washing- University, University of Alberta, University of ton. PIMS is dedicated to: British Columbia, University of Calgary, Univer- sity of Victoria, University of Washington) and af- Promoting innovation and excellence in research filiated Institutions (University of Lethbridge and in all areas encompassed by the mathematical sci- University of Northern British Columbia). ences; The Government of British Columbia through the Initiating collaborations and strengthening ties be- Ministry of Competition, Science and Enterprise, tween the mathematical scientists in the academic The Government of Alberta through the Alberta community and those in the industrial, business and government sectors; Ministry of Innovation and Science, and The Gov- ernment of Canada through the Natural Sciences Training highly qualified personnel for academic and Engineering Research Council of Canada. and industrial employment and creating new op- portunities for developing scientists; Over 350 scientists in its member universities who are actively working towards the Institute’s Developing new technologies to support research, mandate. Their disciplines include pure and ap- communication and training in the mathematical plied mathematics, statistics, computer science, sciences. physical, chemical and life sciences, medical sci- Building on the strength and vitality of its pro- ence, finance, management, and several engineer- grammes, PIMS is able to serve the mathematical ing fields. -
NEWSLETTER Issue: 483 - July 2019
i “NLMS_483” — 2019/6/21 — 14:12 — page 1 — #1 i i i NEWSLETTER Issue: 483 - July 2019 50 YEARS OF MATHEMATICS EXACT AND THERMODYNAMIC AND MINE APPROXIMATE FORMALISM DETECTION COMPUTATIONS i i i i i “NLMS_483” — 2019/6/21 — 14:12 — page 2 — #2 i i i EDITOR-IN-CHIEF COPYRIGHT NOTICE Iain Moatt (Royal Holloway, University of London) News items and notices in the Newsletter may [email protected] be freely used elsewhere unless otherwise stated, although attribution is requested when reproducing whole articles. Contributions to EDITORIAL BOARD the Newsletter are made under a non-exclusive June Barrow-Green (Open University) licence; please contact the author or photog- Tomasz Brzezinski (Swansea University) rapher for the rights to reproduce. The LMS Lucia Di Vizio (CNRS) cannot accept responsibility for the accuracy of Jonathan Fraser (University of St Andrews) information in the Newsletter. Views expressed Jelena Grbic´ (University of Southampton) do not necessarily represent the views or policy Thomas Hudson (University of Warwick) of the Editorial Team or London Mathematical Stephen Huggett (University of Plymouth) Society. Adam Johansen (University of Warwick) Bill Lionheart (University of Manchester) ISSN: 2516-3841 (Print) Mark McCartney (Ulster University) ISSN: 2516-385X (Online) Kitty Meeks (University of Glasgow) DOI: 10.1112/NLMS Vicky Neale (University of Oxford) Susan Oakes (London Mathematical Society) Andrew Wade (Durham University) NEWSLETTER WEBSITE Early Career Content Editor: Vicky Neale The Newsletter is freely available electronically News Editor: Susan Oakes at lms.ac.uk/publications/lms-newsletter. Reviews Editor: Mark McCartney MEMBERSHIP CORRESPONDENTS AND STAFF Joining the LMS is a straightforward process.