DOCSLIB.ORG

Marcelo Viana Director, Instituto Nacional De Mathematica Pure E Aplicada

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Marcelo Viana Director, Instituto Nacional De Mathematica Pure E Aplicada Center for Applied Mathematical Sciences Distinguished Lecturer, Spring 2018 Joint with the Mathematics Department Colloquium Marcelo Viana Director, Instituto Nacional de Mathematica Pure e Aplicada Random products of matrices Abstract: By an old theorem of H. Furstenberg and H. Kesten, the norm of a random product of d-by-d invertible matrices grows at a well-defined (i.e. almost certain) exponential rate, that we call the Lyapunov exponent. A recent result of A. Avila, A. Eskin and myself asserts that this number depends continuously on the data, that is, on the matrix coefficients and their probability weights. For d=2 this was proven before, in my student C. Bocker´s thesis. This behavior is in sharp contrast with some classical results of R. Mane and J. Bochi about the Lyapunov exponents of continuous linear cocycles. I´ll also discuss some moduli of continuity for the Lyapunov exponent of random matrices. Wednesday, January 17, 2018 Marcelo Viana’s research concerns dynamical systems. He has received many honors and awards. In 1998 he was awarded the Prize from the Kaprielian Hall World Academy of Sciences which recognizes excellence in research in the global South. In 2005 he was awarded the inaugural Ramanujan Prize Tea: 3:00 p.m. by ICTP for his research achievements. KAP 410 He is a recipient of Brazil’s National Order of Scientific Merit. In 2016 he was the joint recipient of the Le Grand Lecture: 3:30 p.m. Prix Scientific from the Institut de France. KAP 414 Viana was the Vice-President of the International Mathematical Union ( 2011-2014) and the President of the Brazilian Mathematical society Wine & Cheese: 4:30 p.m. (2013-2015 ). He is Chair of the executive KAP 410 committee for the 2018 International Congress of Mathematicians in Rio de Janeiro. CAMS Director: Susan Friedlander [email protected] .
Load more

Recommended publications
  • 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.
    [Show full text]
  • Continuity of the Lyapunov Exponents of Linear Cocycles
    Continuity of the Lyapunov Exponents of Linear Cocycles Publicações Matemáticas Continuity of the Lyapunov Exponents of Linear Cocycles Pedro Duarte Universidade de Lisboa Silvius Klein PUC-Rio 31o Colóquio Brasileiro de Matemática Copyright 2017 by Pedro Duarte e Silvius Klein Direitos reservados, 2017 pela Associação Instituto Nacional de Matemática Pura e Aplicada - IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ Impresso no Brasil / Printed in Brazil Capa: Noni Geiger / Sérgio R. Vaz 31o Colóquio Brasileiro de Matemática Álgebra e Geometria no Cálculo de Estrutura Molecular - C. Lavor, N. Maculan, M. Souza e R. Alves Continuity of the Lyapunov Exponents of Linear Cocycles - Pedro Duarte e Silvius Klein Estimativas de Área, Raio e Curvatura para H-superfícies em Variedades Riemannianas de Dimensão Três - William H. Meeks III e Álvaro K. Ramos Introdução aos Escoamentos Compressíveis - José da Rocha Miranda Pontes, Norberto Mangiavacchi e Gustavo Rabello dos Anjos Introdução Matemática à Dinâmica de Fluídos Geofísicos - Breno Raphaldini, Carlos F.M. Raupp e Pedro Leite da Silva Dias Limit Cycles, Abelian Integral and Hilbert’s Sixteenth Problem - Marco Uribe e Hossein Movasati Regularization by Noise in Ordinary and Partial Differential Equations - Christian Olivera Topological Methods in the Quest for Periodic Orbits - Joa Weber Uma Breve Introdução à Matemática da Mecânica Quântica - Artur O. Lopes Distribuição: IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ e-mail: [email protected] http://www.impa.br ISBN: 978-85-244-0433-7 i \notes" | 2017/5/29 | 19:08 | page i | #1 i i i Contents Preface 1 1 Linear Cocycles 7 1.1 The definition and examples of ergodic systems .
    [Show full text]
  • Front Matter
    Cambridge University Press 978-1-107-12696-1 - Foundations of Ergodic Theory Marcelo Viana and Krerley Oliveira Frontmatter More information CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 151 Editorial Board B. BOLLOBAS,´ W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO Foundations of Ergodic Theory Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory suitable for a variety of one- or two-semester courses. The authors’ clear and fluent exposition helps the reader to grasp quickly the most important ideas of the theory, and their use of concrete examples illustrates these ideas and puts the results into perspective. The book requires few prerequisites, with background material supplied in the appendix. The first four chapters cover elementary material suitable for undergraduate students – invariance, recurrence and ergodicity – as well as some of the main examples. The authors then gradually build up to more sophisticated topics, including correlations, equivalent systems, entropy, the variational principle, and thermodynamic formalism. The 400 exercises increase in difficulty through the text and test the reader’s understanding of the whole theory. Hints and solutions are provided at the end of the book. Marcelo Viana is Professor of Mathematics at Instituto Nacional de Matematica´ Pura e Aplicada (IMPA), Rio de Janeiro and a leading research expert in ergodic theory and dynamical systems. He has served in several academic organizations, such as the International Mathematical Union (Vice-president 2011–2014), the Brazilian Mathematical Society (President 2013–2015), the Latin American Mathematical Union (Scientific Coordinator, 2001–2008) and the newly founded Mathematical Council of the Americas.
    [Show full text]
  • What Are Lyapunov Exponents, and Why Are They Interesting?
    What are Lyapunov exponents, and why are they interesting? Amie Wilkinson December 6, 2015 Introduction Taking as inspiration the Fields Medal work of Artur Avila, I'd like to introduce you to Lyapunov exponents. My plan is to show how Lyapunov exponents play a key role in three areas in which Avila's results lie: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odingeroperators. But first, what are Lyapunov exponents? Let's begin by viewing them in one of their natural habitats. The barycentric subdivision of a triangle is a collection of 6 smaller triangles obtained by joining the midpoints of the sides to opposite vertices. Here's what happens when you start with an equilateral triangle and repeatedly barycentrically subdivide: Figure 1: Iterated barycentric subdivision, from [McM]. 1 As the subdivision gets successively finer, notice that many of the tri- angles produced by subdivision get increasingly eccentric and needle-like. We can measure the skinniness of a triangle T via the aspect ratio α(T ) = area(T )=L(T )2, where L(T ) is the length of the long side. Suppose we la- bel the triangles in a possible subdivision 1 through 6, roll a six-sided die and at each stage choose a triangle to subdivide. The sequence of triangles T1 ⊃ T2 ⊃ ::: obtained have aspect ratios α1; α2;:::, where αn = α(Tn). Theorem 0.1. There exists a real number χ < 0 such that almost surely: 1 lim log αn = χ. n!1 n In other words, there is a universal constant χ < 0, such that if triangles are chosen successively by a random coin toss, then with probability 1, their aspect ratios will tend to 0 at an exponential rate governed by exp(nχ).
    [Show full text]
  • Joint Journals Catalogue EMS / MSP 2018
    Joint Journals Cataloguemsp EMS / MSP 1 2018 Super package deal inside! & PDE ANALYSIS Volume 9 No. 1 2016 msp msp 1 EEuropean Mathematical Society Mathematical Science Publishers msp 1 The EMS Publishing House is a not-for-profit Mathematical Sciences Publishers is a California organization dedicated to the publication of high- nonprofit corporation based in Berkeley. MSP quality peer-reviewed journals and high-quality honors the best traditions of quality publishing books, on all academic levels and in all fields of while moving with the cutting edge of information pure and applied mathematics. The proceeds from technology. We publish more than 16,000 pages the sale of our publications will be used to keep per year, produce and distribute scientific and the Publishing House on a sound financial footing; research literature of the highest caliber at the any excess funds will be spent in compliance lowest sustainable prices, and provide the top with the purposes of the European Mathematical quality of mathematically literate copyediting and Society. The prices of our products will be set as typesetting in the industry. low as is practicable in the light of our mission and We believe scientific publishing should be an market conditions. industry that helps rather than hinders scholarly activity. High-quality research demands high- Contact addresses quality communication – widely, rapidly and easily European Mathematical Society Publishing House accessible to all – and MSP works to facilitate it. Seminar for Applied Mathematics ETH-Zentrum SEW A21 Contact addresses CH-8092 Zürich, Switzerland Mathematical Sciences Publishers Email: [email protected] 798 Evans Hall #3840 Web: www.ems-ph.org c/o University of California Berkeley, CA 94720-3840, USA Director: Email: [email protected] Dr.
    [Show full text]
  • (I) : Volume in Honor of Jacob Palis - Preliminary Pages
    Astérisque WELLINGTON DE MELO MARCELO VIANA JEAN-CHRISTOPHE YOCCOZ Geometric methods in dynamics (I) : Volume in honor of Jacob Palis - Preliminary pages Astérisque, tome 286 (2003), p. I-XXVIII <http://www.numdam.org/item?id=AST_2003__286__R1_0> © Société mathématique de France, 2003, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ASTÉRISQUE 286 GEOMETRIC METHODS IN DYNAMICS (I) VOLUME IN HONOR OF JACOB PALIS edited by Welington de Melo Marcelo Viana Jean-Christophe Yoccoz Société Mathématique de France 2003 Publié avec le concours du Centre National de la Recherche Scientifique W. de Melo IMPA, Estrada Dona Castorina, 110, Jardim Botânico, Rio de Janeiro 22460-320, Brazil. E-mail : [email protected] Url : www.impa.br/~demelo M. Viana IMPA, Estrada Dona Castorina, 110, Jardim Botânico, Rio de Janeiro 22460-320, Brazil. E-mail : viana0impa.br Url : www.impa.br/~viana J.-C. Yoccoz Collège de France, 11, Place Marcelin Berthelot, 75005 Paris, France. E-mail : [email protected] Url : www.college-de-france.fr/site/equ_dif/p999000715275.htm 2000 Mathematics Subject Classification. — 37-XX, 34-XX, 60-XX, 35-XX. Key words and phrases. — Dynamical systems, ergodic theory, bifurcation theory, dif­ ferential equations.
    [Show full text]
  • The I Nternationa I Congress of Mathematicians
    ____ i 1 Proceedings of ~------ the I nternationa I Congress of Mathematicians August 3-ll, 1994 ZUrich, Switzerland Birkhauser Verlag Basel · Boston · Berlin Editor: S. D. Chatterji EPFL Departement de Mathematiques I 015 Lausanne Switzerland The logo for ICM 94 was designed by Georg Staehelin, Bachweg 6, 8913 Ottenbach, Switzerland. A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data International Congress of Mathematicians <1994, Ziirich>: Proceedings of the International Congress of Mathematicians 1994: August 3- I I, 1994, Zurich, Switzerland I [Ed.: S.D. Chatterji]. - Basel ; Boston ; Berlin : Birkhauser. ISBN 978-3-0348-9897-3 ISBN 978-3-0348-9078-6 (eBook) DOI 10.1007/978-3-0348-9078-6 NE: S. D. Chauerji [Hrsg.] Vol. I ( 1995) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations. broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 1995 B irkhauser Verlag Softcover reprint of the hardcover 1st edition 1995 P.O. Box 133 CH-4010 Basel Switzerland Printed on acid-free paper produced from chlorine-free pulp. TCF oo Layout, typesetting by mathScreen online, Allschwil 98765432 1 Table of Contents Volume I Preface vii Past Congresses viii Past Fields Medalists and Rolf Nevanlinna Prize Winners ix Organization of the Congress . xi The Organizing Committee of the Congress .
    [Show full text]
  • Brazil Report
    BRAZILIAN2 0 MATHEMATICS 18 Contents 02 Brief history Mathematical research 04 10 Graduate studies Brazilian Mathematical Society 12 16 Math Olympiads Math education 18 20 Regional and international cooperation Women in Mathematics 22 24 Math popularization and outreach Brief history 4 Brazil is a relative newcomer to the world of A turning point at the national level was the science, largely due to the late development creation, in 1951, of the two major federal of its institutions of higher education and research agencies, CNPq and CAPES. The research. Early progress, around the beginning Institute for Pure and Applied Mathematics of the 20th century, focused on specifi c (IMPA) was founded by CNPq in 1952, and fi elds such as public health or agriculture. Brazil joined the International Mathematical Mathematics was no priority at that stage. Union (IMU) in 1954. The Brazilian Mathematics Colloquium was created by IMPA in 1957 and BRAZILIAN MATHEMATICS 2018 MATHEMATICS BRAZILIAN The fi rst mathematics seminar was organized has been organized biennially ever since. in 1935 by the Faculty of Philosophy, Sciences Much of Brazilian mathematics grew around it. and Letters of the University of São Paulo. The Faculty had been founded in the previous year and also launched a mathematics journal. In the 1940’s and 1950’s, this institution hired in visiting positions several distinguished foreign mathematicians, including André Weil, Oscar Zariski, Jean Dieudonné and Alexander Grothendieck. LEOPOLDO N ACHBIN: FIRST B RAZILIAN MATHEMATICIAN TO ADDRESS THE I NTERNATIONAL C ONGRESS OF MATHEMATICIANS, IN 1962. GROUP PHOTO FOR THE FIRST B RAZILIAN M ATHEMATICS C OLLOQUIUM, HELD IN P OÇOS DE C ALDAS, STATE OF M INAS G ERAIS, ON J ULY 1-20, 1957.
    [Show full text]
  • Contemporary Mathematics 389
    CONTEMPORARY MATHEMATICS 389 , APORTACIONES MATEMATICAS SOCIEDAD MATEMATICA MEXICANA Geometry and Dynamics International Conference in Honor of the 60th Anniversary of Alberto Verjovsky January 6-ll, 2003 Cuernavaca, Mexico James Eells Etienne Ghys Mikhail Lyubich Jacob Polis Jose Seade Editors http://dx.doi.org/10.1090/conm/389 Geometry and Dynamics CoNTEMPORARY MATHEMATICS 389 ~APORTACIONES MATEMATICAS 'VSOCIEDAD MATEMATICA MEXICANA Geometry and Dynamics International Conference in Honor of the 60th Anniversary of Alberto Verjovsky January 6-11, 2003 Cuernavaca, Mexico James Eells Etienne Ghys Mikhail Lyubich Jacob Polis Jose Seade Editors American Mathematical Society Providence, Rhode Island Editorial Board of Contemporary Mathematics Dennis· DeThrck, managing editor George Andrews Carlos Berenstein Andreas Blass ·Abel Klein Edit.orial Board of Aportaciones Matematicas Luis Gorostiza and Luz De Teresa, Managing Editors Marcelo Aguilar Guillermo Pastor Jose Seade Jose M. Gonzalez-Barrios Raul Quiroga Martha Takane Max Neumann Sergio Rajsbaum Jorge X. Velasco This volume contains the proceedings of the International Conference held January 6-11, 2003 in Cmirnavaca, Mexico, in honor of the 60th anniversary of Alberto Verjovsky. 2000 Mathematics Subject Classification. Primary 17B37, 20Fxx, 32V10, 34Mxx, 37E05, 37Fxx, 53C10, 53D25, 57M30. Library of Congress Cataloging-in-Publication Data Geometry and dynamics : international conference in honor of the 60th anniversary of Alberto Verjovsky, Cuernavaca, Mexico, January 6-11, 2003 /James Eells ... [et al.], editors. p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 389) Includes bibliographical references. ISBN 0-8218-3851-2 (alk. paper) 1. Nonassociative algebras-Congresses. 2. Functions of complex variables-Congresses. 3. Differentiable dynamical systems-Congresses. 4. Geometry, Differential-Congresses. 5. Manifolds (Mathematics)-Congresses.
    [Show full text]
  • First Palis-Balzan Symposium on Dynamical Systems IMPA, from June 25Th Until 29Th, 2012
    First Palis-Balzan Symposium on Dynamical Systems IMPA, from June 25th until 29th, 2012 This symposium was part of the Project Palis-Balzan - Dynamical Systems, Chaotic Behaviour-Uncertainty, sponsored by the Balzan Foundation, since the award conferred by the Balzan Foundation in 2010 to Jacob Palis and by IMPA, supported by CAPES, CNPq and FAPERJ. The project involves scientists from different regions of the world and lasts for five years. Its main goal is to advance the global conjecture on the finiteness of the number of attractors for typical dynamical systems. Some other important topics are: linear cocycles and Lyapunov exponents. The coordinators of this project are Jacob Palis, and Jean-Christophe Yoccoz, 1994 - Fields Medal Winner. As participants there are about 20 other renowned mathematicians, from Brazil (Artur Avila, Carlos Gustavo Moreira, Enrique Pujals, Lorenzo Diaz, Marcelo Viana, Jose Maria Pacífico, Welington de Melo), from France (Carlos Matheus, Sylvain Crovisier, Christian Bonatti and Pierre Berger), from U.S. (Marco Martens, Michael Lyubich), from England (Sebastian van Strien), from China (Lan Wen) and from Uruguay (Martin Sambarino and Alvaro Rovella), and about 10 young Ph.Ds. This symposium aims to promote research at the highest level in the dynamical systems area, especially in the topics cited above, with the effective participation of excellent national and foreign researchers. It also aims to put the doctoral students and young researchers in contact with the best that is produced worldwide in the issues mentioned above and related ones, disseminating recent results and providing an international scientific exchange at the highest level. It stimulated the development of the Brazilian group in the area, which is increasingly asserting itself in the global context as one of the leading players.
    [Show full text]
  • December 2003
    THE LONDON MATHEMATICAL SOCIETY NEWSLETTER No. 321 December 2003 Forthcoming COUNCIL DIARY The Publisher had attended a 17 October 2003 workshop on Scenario Planning Society in Publishing, and reported to Meetings Already before the October Council on the issues under dis- Council meeting, members of cussion. There is a need for 2004 Council had been circulated some long term planning to Friday 9 January urgently by email for their deal with long term threats to AMS Meeting, individual responses to the publishing. The Society’s pub- Arizona HEFCE consultation on devel- lishing activities represent both G. van der Geer oping the funding method for a service to the mathematical [page X] teaching in English universi- community and a source of ties from 2004-5. The propos- income which together with Friday 20 February als suggest a 5.4% cut in the our investments funds our 1 London mathematics teaching budg- other activities; we need to D. Schleicher et. Council approved the maintain their health. S.M. Rees excellent document which the As Programme Secretary, (Mary Cartwright Education Secretary had sub- Stephen Huggett reported on Lecture) sequently put together. It has the impact that budget cuts are [page X] necessarily been a fast having on Programme response; it is now important Committee grants. Programme Wednesday 12 May that we find common cause Committee’s policy is to aim to Nottingham with as many others as possi- fund the parts which other Midlands Regional ble, that we target MPs, and grants cannot reach, in other Meeting talk to the Press. words, it tries to complement In this climate, the announce- other grant awarding bodies in Friday 18 June ment from the Treasurer that, the way it directs its funding.
    [Show full text]
  • Of the American Mathematical Society March 2013 Volume 60, Number 3
    ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society March 2013 Volume 60, Number 3 Remembering Walter Rudin (1921–2010) page 295 Cancer Modeling: A Personal Perspective page 304 A Revolutionary Material page 310 Simulating the development of a brain tumor (see page 325) Open Access Journals Abstract and in Mathematics Applied Analysis Your research wants to be free! Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Advances in Decision Sciences Advances in Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation Volume 2013 http://www.hindawi.com 7 6 Advances in 9 15 Operations Algebra 768 Research 7 Advances in Submit your manuscripts at Numerical Analysis Hindawi Publishing Corporation3 Volume 2013 Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com 2 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 Discrete Dynamics in Nature and Society Computational and Mathematical Methods International Journal of in Medicine Game Theory Geometry Analysis Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 International Journal of Mathematics and Mathematical Sciences International Journal
    [Show full text]