ON the ISOMORPHISM PROBLEM for NON-MINIMAL TRANSFORMATIONS with DISCRETE SPECTRUM Nikolai Edeko (Communicated by Bryna Kra) 1. I
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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. -
Quasi-Invariant Gaussian Measures for the Cubic Fourth Order Nonlinear Schrödinger Equation
Probab. Theory Relat. Fields DOI 10.1007/s00440-016-0748-7 Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation Tadahiro Oh1,2 · Nikolay Tzvetkov3 Received: 4 August 2015 / Revised: 25 November 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We consider the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces s(T) > 3 H , s 4 , are quasi-invariant under the flow. Keywords Fourth order nonlinear Schrödinger equation · Biharmonic nonlinear Schrödinger equation · Gaussian measure · Quasi-invariance Mathematics Subject Classification 35Q55 Contents 1 Introduction ............................................... 1.1 Background ............................................. 1.2 Cubic fourth order nonlinear Schrödinger equation ........................ 1.3 Main result ............................................. 1.4 Organization of the paper ...................................... 2 Notations ................................................. 3 Reformulation of the cubic fourth order NLS .............................. B Tadahiro Oh [email protected] Nikolay Tzvetkov [email protected] 1 School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK 2 The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, -
I. Overview of Activities, April, 2005-March, 2006 …
MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 …......……………………. 2 Innovations ………………………………………………………..... 2 Scientific Highlights …..…………………………………………… 4 MSRI Experiences ….……………………………………………… 6 II. Programs …………………………………………………………………….. 13 III. Workshops ……………………………………………………………………. 17 IV. Postdoctoral Fellows …………………………………………………………. 19 Papers by Postdoctoral Fellows …………………………………… 21 V. Mathematics Education and Awareness …...………………………………. 23 VI. Industrial Participation ...…………………………………………………… 26 VII. Future Programs …………………………………………………………….. 28 VIII. Collaborations ………………………………………………………………… 30 IX. Papers Reported by Members ………………………………………………. 35 X. Appendix - Final Reports ……………………………………………………. 45 Programs Workshops Summer Graduate Workshops MSRI Network Conferences MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 This annual report covers MSRI projects and activities that have been concluded since the submission of the last report in May, 2005. This includes the Spring, 2005 semester programs, the 2005 summer graduate workshops, the Fall, 2005 programs and the January and February workshops of Spring, 2006. This report does not contain fiscal or demographic data. Those data will be submitted in the Fall, 2006 final report covering the completed fiscal 2006 year, based on audited financial reports. This report begins with a discussion of MSRI innovations undertaken this year, followed by highlights -
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. -
Boshernitzan's Condition, Factor Complexity, and an Application
BOSHERNITZAN’S CONDITION, FACTOR COMPLEXITY, AND AN APPLICATION VAN CYR AND BRYNA KRA Abstract. Boshernitzan found a decay condition on the measure of cylin- der sets that implies unique ergodicity for minimal subshifts. Interest in the properties of subshifts satisfying this condition has grown recently, due to a connection with the study of discrete Schr¨odinger operators. Of particular interest is the question of how restrictive Boshernitzan’s condition is. While it implies zero topological entropy, our main theorem shows how to construct minimal subshifts satisfying the condition whose factor complexity grows faster than any pre-assigned subexponential rate. As an application, via a theorem of Damanik and Lenz, we show that there is no subexponentially growing se- quence for which the spectra of all discrete Schr¨odinger operators associated with subshifts whose complexity grows faster than the given sequence, have only finitely many gaps. 1. Boshernitzan’s complexity conditions For a symbolic dynamical system (X, σ), many of the isomorphism invariants we have are statements about the growth rate of the word complexity function PX (n), which counts the number of distinct cylinder sets determined by words of length n having nonempty intersection with X. For example, the exponential growth of PX (n) is the topological entropy of (X, σ), while the linear growth rate of PX (n) gives an invariant to begin distinguishing between zero entropy systems. Of course there are different senses in which the growth of PX (n) could be said to be linear and different invariants arise from them. For example, one can consider systems with linear limit inferior growth, meaning lim infn→∞ PX (n)/n < ∞, or the stronger condition of linear limit superior growth, meaning lim supn→∞ PX (n)/n < ∞. -
Attractors and Orbit-Flip Homoclinic Orbits for Star Flows
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 8, August 2013, Pages 2783–2791 S 0002-9939(2013)11535-2 Article electronically published on April 12, 2013 ATTRACTORS AND ORBIT-FLIP HOMOCLINIC ORBITS FOR STAR FLOWS C. A. MORALES (Communicated by Bryna Kra) Abstract. We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be C1 approximated by vector fields with orbit-flip homoclinic orbits. 1. Introduction The notion of attractor deserves a fundamental place in the modern theory of dynamical systems. This assertion, supported by the nowadays classical theory of turbulence [27], is enlightened by the recent Palis conjecture [24] about the abun- dance of dynamical systems with finitely many attractors absorbing most positive trajectories. If confirmed, such a conjecture means the understanding of a great part of dynamical systems in the sense of their long-term behaviour. Here we attack a problem which goes straight to the Palis conjecture: The fini- tude of the number of attractors for a given dynamical system. Such a problem has been solved positively under certain circunstances. For instance, we have the work by Lopes [16], who, based upon early works by Ma˜n´e [18] and extending pre- vious ones by Liao [15] and Pliss [25], studied the structure of the C1 structural stable diffeomorphisms and proved the finitude of attractors for such diffeomor- phisms. His work was largely extended by Ma˜n´e himself in the celebrated solution of the C1 stability conjecture [17]. On the other hand, the Japanesse researchers S. -
A Brief on Characteristic Functions
Missouri University of Science and Technology Scholars' Mine Graduate Student Research & Creative Works Student Research & Creative Works 01 Dec 2020 A Brief on Characteristic Functions Austin G. Vandegriffe Follow this and additional works at: https://scholarsmine.mst.edu/gradstudent_works Part of the Applied Mathematics Commons, and the Probability Commons Recommended Citation Vandegriffe, Austin G., "A Brief on Characteristic Functions" (2020). Graduate Student Research & Creative Works. 2. https://scholarsmine.mst.edu/gradstudent_works/2 This Presentation is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Graduate Student Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. A Brief on Characteristic Functions A Presentation for Harmonic Analysis Missouri S&T : Rolla, MO Presentation by Austin G. Vandegriffe 2020 Contents 1 Basic Properites of Characteristic Functions 1 2 Inversion Formula 5 3 Convergence & Continuity of Characteristic Functions 9 4 Convolution of Measures 13 Appendix A Topology 17 B Measure Theory 17 B.1 Basic Measure Theory . 17 B.2 Convergence in Measure & Its Consequences . 20 B.3 Derivatives of Measures . 22 C Analysis 25 i Notation 8 For all 8P For P-almost all, where P is a measure 9 There exists () If and only if U Disjoint union -
Nilpotent Structures in Ergodic Theory
Mathematical Surveys and Monographs Volume 236 Nilpotent Structures in Ergodic Theory Bernard Host Bryna Kra 10.1090/surv/236 Nilpotent Structures in Ergodic Theory Mathematical Surveys and Monographs Volume 236 Nilpotent Structures in Ergodic Theory Bernard Host Bryna Kra EDITORIAL COMMITTEE Walter Craig Natasa Sesum Robert Guralnick, Chair Benjamin Sudakov Constantin Teleman 2010 Mathematics Subject Classification. Primary 37A05, 37A30, 37A45, 37A25, 37B05, 37B20, 11B25,11B30, 28D05, 47A35. For additional information and updates on this book, visit www.ams.org/bookpages/surv-236 Library of Congress Cataloging-in-Publication Data Names: Host, B. (Bernard), author. | Kra, Bryna, 1966– author. Title: Nilpotent structures in ergodic theory / Bernard Host, Bryna Kra. Description: Providence, Rhode Island : American Mathematical Society [2018] | Series: Mathe- matical surveys and monographs; volume 236 | Includes bibliographical references and index. Identifiers: LCCN 2018043934 | ISBN 9781470447809 (alk. paper) Subjects: LCSH: Ergodic theory. | Nilpotent groups. | Isomorphisms (Mathematics) | AMS: Dynamical systems and ergodic theory – Ergodic theory – Measure-preserving transformations. msc | Dynamical systems and ergodic theory – Ergodic theory – Ergodic theorems, spectral theory, Markov operators. msc | Dynamical systems and ergodic theory – Ergodic theory – Relations with number theory and harmonic analysis. msc | Dynamical systems and ergodic theory – Ergodic theory – Ergodicity, mixing, rates of mixing. msc | Dynamical systems and ergodic -
Sarnak's Möbius Disjointness for Dynamical Systems with Singular Spectrum and Dissection of Möbius Flow
Sarnak’s Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow Houcein Abdalaoui, Mahesh Nerurkar To cite this version: Houcein Abdalaoui, Mahesh Nerurkar. Sarnak’s Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow. 2020. hal-02867657 HAL Id: hal-02867657 https://hal.archives-ouvertes.fr/hal-02867657 Preprint submitted on 15 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sarnak’s M¨obius disjointness for dynamical systems with singular spectrum and dissection of M¨obius flow El Houcein El Abdalaoui Department of Mathematics Universite of Rouen, France and Mahesh Nerurkar Department of Mathematics Rutgers University, Camden NJ 08102 USA Abstract It is shown that Sarnak’s M¨obius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is accomplished by first establishing a special case of Chowla conjecture which gives a correlation between the M¨obius function and its square. Then a computation of W. Veech, followed by an argument using the notion of ‘affinity between measures’, (or the so called ‘Hellinger method’), completes the proof. -
Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems Elise Janvresse, Emmanuel Roy, Thierry De La Rue
Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems Elise Janvresse, Emmanuel Roy, Thierry de la Rue To cite this version: Elise Janvresse, Emmanuel Roy, Thierry de la Rue. Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems. Encyclopedia of Complexity and Systems Science, 2020, 978-3-642- 27737-5. 10.1007/978-3-642-27737-5_725-1. hal-02451104 HAL Id: hal-02451104 https://hal.archives-ouvertes.fr/hal-02451104 Submitted on 23 Jan 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. DYNAMICAL SYSTEMS OF PROBABILISTIC ORIGIN: GAUSSIAN AND POISSON SYSTEMS ELISE´ JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE Outline Glossary 1 1. Definition of the subject 3 2. Introduction 4 3. From probabilistic objects to dynamical systems 4 3.1. Gaussian systems 4 3.2. Poisson suspensions 5 4. Spectral theory 6 4.1. Basics of spectral theory 6 4.2. Fock space 7 4.3. Operators on a Fock space 7 4.4. Application to Gaussian and Poisson chaos 7 5. Basic ergodic properties 8 5.1. Ergodicity and mixing 8 5.2. Entropy, Bernoulli properties 9 6. Joinings, factors and centralizer 10 6.1. -
2021 September-October Newsletter
Newsletter VOLUME 51, NO. 5 • SEPTEMBER–OCTOBER 2021 PRESIDENT’S REPORT This is a fun report to write, where I can share news of AWM’s recent award recognitions. Sometimes hearing about the accomplishments of others can make The purpose of the Association for Women in Mathematics is us feel like we are not good enough. I hope that we can instead feel inspired by the work these people have produced and energized to continue the good work we • to encourage women and girls to ourselves are doing. study and to have active careers in the mathematical sciences, and We’ve honored exemplary Student Chapters. Virginia Tech received the • to promote equal opportunity and award for Scientific Achievement for offering three different research-focused the equal treatment of women and programs during a pandemic year. UC San Diego received the award for Professional girls in the mathematical sciences. Development for offering multiple events related to recruitment and success in the mathematical sciences. Kutztown University received the award for Com- munity Engagement for a series of events making math accessible to a broad community. Finally, Rutgers University received the Fundraising award for their creative fundraising ideas. Congratulations to all your members! AWM is grateful for your work to support our mission. The AWM Research Awards honor excellence in specific research areas. Yaiza Canzani was selected for the AWM-Sadosky Research Prize in Analysis for her work in spectral geometry. Jennifer Balakrishnan was selected for the AWM- Microsoft Research Prize in Algebra and Number Theory for her work in computa- tional number theory. -
DMS COV Report
REPORT OF THE 2013 COMMITTEE OF VISITORS DIVISION OF MATHEMATICAL SCIENCES NATIONAL SCIENCE FOUNDATION February 20-22, 2013 Submitted on behalf of the Committee by Mark L. Green, Chair to Fleming Crim Assistant Director Mathematics and Physical Sciences The mathematical sciences constitute a rich and complex ecosystem, one in which people and ideas move across boundaries. It is constantly evolving. For the mathematical sciences to be healthy, all parts of this ecosystem must be nurtured. This is a period of rapid change in the mathematical landscape. As detailed in “The Mathematical Sciences in 2025,” there has been a notable expansion in the impact of the mathematical sciences on other subjects, and in the types of mathematical and statistical ideas being used. It has been a Golden Age for core mathematical sciences, with major advances on longstanding problems and innovation in fundamental theory. It is also a time of economic strains on universities, with potentially quite significant implications for the mathematical sciences. These challenges and opportunities, as they unfold, will require continued evolution in how the mathematical sciences are funded. The Committee of Visitors’ fundamental mission was to assess the integrity and efficiency of the proposal review process in the Division of Mathematical Sciences (DMS) for the period 2010-2012. DMS was commendably forthcoming with the information that we needed for this study, even when it was hard to get, both in overall statistical information and in aspects specific to a program, panel or proposal. The Committee of Visitors was satisfied that the system in place functions very well overall. We found the quality and significance of the division’s programmatic investments to be extremely high.