IAMP News Bulletin July 2018
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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 . -
Joaquin M. Luttinger 1923–1997
Joaquin M. Luttinger 1923–1997 A Biographical Memoir by Walter Kohn ©2014 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. JOAQUIN MAZDAK LUTTINGER December 2, 1923–April 6, 1997 Elected to the NAS, 1976 The brilliant mathematical and theoretical physicist Joaquin M. Luttinger died at the age of 73 years in the city of his birth, New York, which he deeply loved throughout his life. He had been in good spirits a few days earlier when he said to Walter Kohn (WK), his longtime collaborator and friend, that he was dying a happy man thanks to the loving care during his last illness by his former wife, Abigail Thomas, and by his stepdaughter, Jennifer Waddell. Luttinger’s work was marked by his exceptional ability to illuminate physical properties and phenomena through Visual Archives. Emilio Segrè Photograph courtesy the use of appropriate and beautiful mathematics. His writings and lectures were widely appreciated for their clarity and fine literary quality. With Luttinger’s death, an By Walter Kohn influential voice that helped shape the scientific discourse of his time, especially in condensed-matter physics, was stilled, but many of his ideas live on. For example, his famous 1963 paper on condensed one-dimensional fermion systems, now known as Tomonaga-Luttinger liquids,1, 2 or simply Luttinger liquids, continues to have a strong influence on research on 1-D electronic dynamics. In the 1950s and ’60s, Luttinger also was one of the great figures who helped construct the present canon of classic many-body theory while at the same time laying founda- tions for present-day revisions. -
Arxiv:Cond-Mat/0106256 V1 13 Jun 2001
Quantum Phenomena in Low-Dimensional Systems Michael R. Geller Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602-2451 (June 18, 2001) A brief summary of the physics of low-dimensional quantum systems is given. The material should be accessible to advanced physics undergraduate students. References to recent review articles and books are provided when possible. I. INTRODUCTION transverse eigenfunction and only the plane wave factor changes. This means that motion in those transverse di- A low-dimensional system is one where the motion rections is \frozen out," leaving only motion along the of microscopic degrees-of-freedom, such as electrons, wire. phonons, or photons, is restricted from exploring the full This article will provide a very brief introduction to three dimensions of our world. There has been tremen- the physics of low-dimensional quantum systems. The dous interest in low-dimensional quantum systems dur- material should be accessible to advanced physics under- ing the past twenty years, fueled by a constant stream graduate students. References to recent review articles of striking discoveries and also by the potential for, and and books are provided when possible. The fabrication of realization of, new state-of-the-art electronic device ar- low-dimensional structures is introduced in Section II. In chitectures. Section III some general features of quantum phenomena The paradigm and workhorse of low-dimensional sys- in low dimensions are discussed. The remainder of the tems is the nanometer-scale semiconductor structure, or article is devoted to particular low-dimensional quantum semiconductor \nanostructure," which consists of a com- systems, organized by their \dimension." positionally varying semiconductor alloy engineered at the atomic scale [1]. -
![Arxiv:1609.08664V2 [Math-Ph] 9 Dec 2017 Yhftde N[ in Hofstadter by Ae[ Case Atrpern Sntitne O Publication](https://docslib.b-cdn.net/cover/0393/arxiv-1609-08664v2-math-ph-9-dec-2017-yhftde-n-in-hofstadter-by-ae-case-atrpern-sntitne-o-publication-1150393.webp)
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]. -
Chicago Physics One
CHICAGO PHYSICS ONE 3:25 P.M. December 02, 1942 “All of us... knew that with the advent of the chain reaction, the world would never be the same again.” former UChicago physicist Samuel K. Allison Physics at the University of Chicago has a remarkable history. From Albert Michelson, appointed by our first president William Rainey Harper as the founding head of the physics department and subsequently the first American to win a Nobel Prize in the sciences, through the mid-20th century work led by Enrico Fermi, and onto the extraordinary work being done in the department today, the department has been a constant source of imagination, discovery, and scientific transformation. In both its research and its education at all levels, the Department of Physics instantiates the highest aspirations and values of the University of Chicago. Robert J. Zimmer President, University of Chicago Welcome to the inaugural issue of Chicago Physics! We are proud to present the first issue of Chicago Physics – an annual newsletter that we hope will keep you connected with the Department of Physics at the University of Chicago. This newsletter will introduce to you some of our students, postdocs and staff as well as new members of our faculty. We will share with you good news about successes and recognition and also convey the sad news about the passing of members of our community. You will learn about the ongoing research activities in the Department and about events that took place in the previous year. We hope that you will become involved in the upcoming events that will be announced. -
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. -
(Oxon) on the THEORY of MODERN QUANTUM ALGORITHMS
SKOLKOVO INSTITUTE OF SCIENCE AND TECHNOLOGY draft version Jacob Daniel Biamonte, BS, DPhil (Oxon) ON THE THEORY OF MODERN QUANTUM ALGORITHMS Specialization 01.01.03 Mathematical Physics Dissertation submitted for the degree of Doctor of Physical and Mathematical Sciences arXiv:2009.10088v1 [quant-ph] 21 Sep 2020 Moscow | 2020 2 Contents Page Chapter 1. Introduction: Programming Ground States .......5 1.1 Computation and the Ising model . .9 1.2 Low-energy subspace embedding . 15 1.3 Two-body reductions . 19 1.4 P- vs. NP problems and physics . 23 1.5 Computational phase transitions . 24 1.5.1 Thermal states at the SAT phase transition . 29 Chapter 2. Quantum vs Probabilistic Computation ......... 32 2.1 Defining Mechanics . 32 2.1.1 Stochastic time evolution . 33 2.1.2 Quantum time evolution . 33 2.1.3 Properties in stochastic mechanics . 34 2.1.4 Hamiltonian properties in quantum mechanics . 35 2.1.5 Observables . 36 2.2 Walks on Graphs: quantum vs stochastic . 44 2.2.1 Normalized Laplacians . 47 2.2.2 Stochastic walk . 50 2.2.3 Quantum walk . 51 2.2.4 Perron's theorem . 54 2.3 Google page rank|a ground eigenvector problem . 59 2.4 Kitaev's quantum phase estimation algorithm . 61 2.5 Finding the ground state . 66 Chapter 3. Tensor Networks and Quantum Circuits ......... 69 3.1 Clifford gates . 69 3.2 Tensor network building blocks . 72 3.2.1 Reversible logic . 75 3.2.2 Heisenberg picture . 77 3.2.3 Stabilizer tensor theory . 79 3 Chapter 4. Variational Search and Optimization .......... -
A DIALOGUE on the THEORY of HIGH Tc
A DIALOGUE ON THE THEORY OF HIGH Tc Superconducting Bi2Sr2CaCu2O,, as seen with reflected differential interference contrast microscopy. This view of the ab plane surface of a platelet of the ceramic material shows this high-temperature superconductor's strong planar structure. (Photomicrograph by Michael W. Davidson, W. Jack Rink and Joseph B. Schlenoff, Florida State University.) Figure 1 5 4 PHYSICS TODAY JUNE 1991 The give-and-take between two solid-state theorists offers insight into materials with high superconducting transition temperatures and illustrates the kind of thinking that goes into developing a new theory. Philip W. Anderson and Robert Schrieffer Although ideas that would explain the behavior of the formalism to do their calculations.' I believe they are high-temperature superconducting materials have been wrong. I'd like to hear your opinion, but first let me say a offered almost since their discovery, high-Tt. theory is still couple of things that bear on this question. In the first very much in flux. Two of the leading figures in condensed place, I think few people realize that we now know of at matter theory are Philip Anderson, the Joseph Henry least six different classes of electron superconductors, and Professor of Physics at Princeton University, and Robert two other BCS fluids as well. Out of these only one obeys Schrieffer, Chancellor's Professor at the University of the so-called conventional theory—that is, BCS with California, Santa Barbara. Anderson's ideas have fo- phonons that fit unmodified versions of Eliashberg's cused, in his own words, "on a non-Fermi-liquid normal equations. -
Emergence and Reductionism: an Awkward Baconian Alliance
Emergence and Reductionism: an awkward Baconian alliance Piers Coleman1;3 1Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA and 2 Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK. Abstract This article discusses the relationship between emergence and reductionism from the perspective of a condensed matter physicist. Reductionism and emergence play an intertwined role in the everyday life of the physicist, yet we rarely stop to contemplate their relationship: indeed, the two are often regarded as conflicting world-views of science. I argue that in practice, they compliment one-another, forming an awkward alliance in a fashion envisioned by the philosopher scientist, Francis Bacon. Looking at the historical record in classical and quantum physics, I discuss how emergence fits into a reductionist view of nature. Often, a deep understanding of reductionist physics depends on the understanding of its emergent consequences. Thus the concept of energy was unknown to Newton, Leibniz, Lagrange or Hamilton, because they did not understand heat. Similarly, the understanding of the weak force awaited an understanding of the Meissner effect in superconductivity. Emergence can thus be likened to an encrypted consequence of reductionism. Taking examples from current research, including topological insulators and strange metals, I show that the convection between emergence and reductionism continues to provide a powerful driver for frontier scientific research, linking the lab with the cosmos. Article to be published by Routledge (Oxford and New York) as part of a volume entitled arXiv:1702.06884v2 [physics.hist-ph] 7 Dec 2017 \Handbook of Philosophy of Emergence", editors Sophie Gibb, Robin Hendry and Tom Lancaster.