DOCSLIB.ORG

Annual Report 1999–2000

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Annual Report 1999–2000 Cover Art: The seemingly Rorschach image on the cover was created by a program written by Cornell undergraduate Karl Papadantonakis, under the direction of Professor John Hubbard. This and other programs and information are available at http://www.math.cornell.edu/ dynamics. One of the earliest and most famous functions studied in dynamical systems is the complex quadratic 2 fc : z z + c, where c is a complex parameter. Professor Hubbard signicantly contributed to the → e understanding of fc in the 1980s. His intuition was aided by helpful computer-generated images, like the Mandelbrot set. It has been a long time goal of his to create similar pictures for the more complicated 2 2 Henon map, Ha,c :(x, y) (x + c ay, x), a dieomorphism of C with two complex parameters a, c. This goal is just being realized→ through several new programs by Mr. Papadantonakis. The cover image is associated with a Henon map with a = .2001416015625,c= 1.44404296875. Since 2 2 H is a map of C , the picture shows only a slice of C , but one which is dynamically signicant. Pixels, x0, are colored black if their orbits x0,H(x0),H(H(x0),... remain bounded, and colored according to rate of escape if they diverge to innity.{ It is the goal of Professor} Hubbard, and others at Cornell, to use these new programs to further their understanding of the Henon map. Department of Mathematics Annual Report 1999Ð2000 Year in Review: Mathematics Instruction and Research Cornell University Þrst among private institutions in undergraduates who later earn Ph.D.s. Ithaca, New York, the home of Cornell University, is located in the heart of the Finger Lakes Region. It offers the cultural activities of a large university and the diversions of a rural environment. Mathematics study at Cornell is a unique experience. The university has managed to foster excellence in research without forsaking the ideals of a liberal education. In many ways, the cohesiveness and rigor of the Mathematics Department is a reßection of the Cornell tradition. John Smillie, chair Department of Mathematics Cornell University Telephone: (607) 255-4013 320 Malott Hall Fax: (607) 255-7149 Ithaca, NY 14853-4201 e-mail: [email protected] Table of Contents The Year in Review 1999–2000.......................................................................1 VIGRE ..................................................................................................2 Graduate Program .....................................................................................3 Undergraduate Program ..............................................................................4 Research and Professional Activities.................................................................4 Support Sta ...........................................................................................4 SCREMS Grant ........................................................................................5 Faculty Changes .......................................................................................5 Gifts .....................................................................................................6 Awards and Honors ....................................................................................6 Clark Distinguished Teaching Award; Eleanor Norton York Award; Freshman Math Prize; Goldman Excellence in Teaching Award; Goldwater Scholarship; Hutchinson Fellowship; Ithaca High School Senior Prize; Kieval Prize; Robert John Battig Graduate Prize Instructional Activities ................................................................................7 Curriculum Changes ...................................................................................7 Interdisciplinary Instructional Activity ..............................................................8 Mathematics/Engineering Liaison; Engineering Restructuring New Faculty Members .................................................................................9 Mathematics Course Enrollment Statistics .......................................................10 Mathematics Department Directory 1999–2000 .................................................12 Special Programs and Activities ....................................................................14 Roger Farrell’s Retirement...........................................................................15 Finite Element Circus ................................................................................14 A Midlife Crisis Funded by the NIH................................................................14 Math Club Guest Lectures ..........................................................................15 Summer Program .....................................................................................15 Spring Concert Series ................................................................................16 Topology Festival .....................................................................................16 Preparing Future Professors .........................................................................17 Expanding Your Horizons ...........................................................................17 Mathematics Awareness Month .....................................................................17 Research Experiences for Undergraduates Program ..............................................18 Centers for Mathematics Research at Cornell .....................................................19 Center for Applied Mathematics; Center for the Foundations of Intelligent Systems Mathematics Library .................................................................................19 Special Instructional Support .......................................................................20 Computer Lab; Mathematics Support Center; Learning Strategies Center NSF Undergraduate Faculty Enhancement Workshop............................................22 Mathematics Education..............................................................................22 Cornell/Schools Mathematics Resource Program; Teacher Education in Agriculture, Mathematics and Science Mathematics Department Endowments ............................................................22 The Colloquium Endowment Fund; The Eleanor Norton York Endowment; The Faculty Book Endowment; The Israel Berstein Memorial Fund; The Logic Endowment; The Robert John Battig Endowment Degrees Granted 1999–2000 ........................................................................24 Department Colloquia ................................................................................28 1999–2000 Faculty Publications ....................................................................35 The Faculty and their Research .....................................................................37 Faculty Proles.........................................................................................39 Visiting Faculty Program Participants .............................................................74 Sta Proles............................................................................................77 The Year in Review 1999–2000 This was my rst year as chair. I am happy to re- engineering calculus. We were assisted in teaching Math port that the department seems to have survived. The 191, Math 193 and Math 192 during the fall 1999 semes- department also beneted from the eorts of Professor ter by faculty from throughout the campus, additional Stephen Chase as director of undergraduate studies, Pro- instructional teaching assistants and an additional H. C. fessor Dan Barbasch as director of graduate studies and Wang Assistant Professor. The engineering restructur- Senior Lecturer Thomas Rishel as teaching assistant co- ing eort will continue in its present form for at least ordinator. In addition, Professor Louis Billera served two more years: the fall of 2000 and the fall of 2001. as chair of the computer committee, Professor Robert Strichartz as chair of the mathematics major commit- Our graduate student community was active and pro- tee, Professor Birgit Speh as chair of the faculty search ductive this year, resulting in a variety of honors and committee and Professor John Guckenheimer as chair of awards. Leah Gold gave a lecture at the Special Session the curriculum committee. Professors Cohen and Morley in Syzygies conference at the AMS meeting in Lowell, headed up the undergraduate advising committee (with April 2000. Matthew Horak and Leah Gold received the help of Steve Chase), and Assistant Professor Ravi Hutchinson Fellowships for spring 2000. Lek-Heng Lim Ramakrishna devoted considerable time and energy to was awarded a Clare Hall Fellowship for 2000–01. An- the undergraduate Math Club committee. In addition, tal Jarai received the Battig Prize in fall 1999. Stephen Assistant Professor Reyer Sjamaar did a great job of co- Bullock received the Eleanor Norton York award in fall ordinating the Oliver Club for the 1999–2000 academic 1999. David Brown was awarded the Clark Teaching year. Professor Richard Durrett agreed to ll the newly Award in spring 2000. created positon of VIGRE director. (See VIGRE, p. 2.) Each year we award the Kieval prize to an outstanding We conducted a number of faculty searches during the undergraduate senior. This year it was given to Karl Pa- 1999–2000 academic year. I am happy to report that we padantonakis, who graduated summa cum laude in May were quite successful in lling our vacant H. C. Wang 2000. Karl was a double major in the College Scholar
Load more

Recommended publications
  • Multiple Mixing from Weak Hyperbolicity by the Hopf Argument Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy
    ,pdfcreator=HAL,pdfproducer=PDFLaTeX,pdfsubject=Mathematics [math]/Dynamical Systems [math.DS] Multiple mixing from weak hyperbolicity by the Hopf argument Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy To cite this version: Yves Coudène, Boris Hasselblatt, Serge Troubetzkoy. Multiple mixing from weak hyperbolicity by the Hopf argument. 2014. hal-01006451v2 HAL Id: hal-01006451 https://hal.archives-ouvertes.fr/hal-01006451v2 Submitted on 16 Jun 2014 (v2), last revised 15 Sep 2015 (v3) 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. MULTIPLE MIXING FROM WEAK HYPERBOLICITY BY THE HOPF ARGUMENT YVES COUDÈNE, BORIS HASSELBLATT AND SERGE TROUBETZKOY ABSTRACT. We show that using only weak hyperbolicity (no smoothness, com- pactness or exponential rates) the Hopf argument produces multiple mixing in an elementary way. While this recovers classical results with far simpler proofs, the point is the broader applicability implied by the weak hypothe- ses. Some of the results can also be viewed as establishing “mixing implies multiple mixing” outside the classical hyperbolic context. 1. INTRODUCTION The origins of hyperbolic dynamical systems are connected with the efforts by Boltzmann and Maxwell to lay a foundation under statistical mechanics. In today’s terms their ergodic hypothesis was that the mechanical system defined by molecules in a container is ergodic, and the difficulties of establishing this led to the search for any mechanical systems with this property.
    [Show full text]
  • Diagonalizable Flows on Locally Homogeneous Spaces and Number
    Diagonalizable flows on locally homogeneous spaces and number theory Manfred Einsiedler and Elon Lindenstrauss∗ Abstract.We discuss dynamical properties of actions of diagonalizable groups on locally homogeneous spaces, particularly their invariant measures, and present some number theoretic and spectral applications. Entropy plays a key role in the study of theses invariant measures and in the applications. Mathematics Subject Classification (2000). 37D40, 37A45, 11J13, 81Q50 Keywords. invariant measures, locally homogeneous spaces, Littlewood’s conjecture, quantum unique ergodicity, distribution of periodic orbits, ideal classes, entropy. 1. Introduction Flows on locally homogeneous spaces are a special kind of dynamical systems. The ergodic theory and dynamics of these flows are very rich and interesting, and their study has a long and distinguished history. What is more, this study has found numerous applications throughout mathematics. The spaces we consider are of the form Γ\G where G is a locally compact group and Γ a discrete subgroup of G. Typically one takes G to be either a Lie group, a linear algebraic group over a local field, or a product of such. Any subgroup H < G acts on Γ\G and this action is precisely the type of action we will consider here. One of the most important examples which features in numerous number theoretical applications is the space PGL(n, Z)\ PGL(n, R) which can be identified with the space of lattices in Rn up to homothety. Part of the beauty of the subject is that the study of very concrete actions can have meaningful implications. For example, in the late 1980s G.
    [Show full text]
  • Anatole Katok Center for Dynamical Systems and Geometry Svetlana Katok and Yakov Pesin
    COMMUNICATION Anatole Katok Center for Dynamical Systems and Geometry Svetlana Katok and Yakov Pesin Creation of the Center Yakov Pesin and Howard Weiss in Seattle in 1999 and the The Penn State research group in dynamical systems was International Conference “Ergodic Theory, Geometric Ri- formed in 1990 when Anatole and Svetlana Katok, Yakov gidity, and Number Theory,” which was co-organized by Pesin, and Howard Weiss moved to Penn State to join Eu- Anatole at Newton Institute (Cambridge, UK) in July, 2000. gene Wayne who already was there. By that time Anatole Within a few years after its creation, the dynamics group had already established himself as a leader in dynamical at Penn State had grown and so had its activities. This in- systems who had co-organized several major events (such cluded a visitor program, intensive collaborations, a weekly as a year program in dynamics at MSRI in 1983-84) and seminar, etc. The group also had a large number of graduate had trained a number of graduate students and postdocs at students interested in dynamics. In early 1990, due to the University of Maryland and Caltech. He came to Penn State fall of the Soviet Union, many undergraduates from East- with the plan to build a strong group in dynamics and to ern European countries had the opportunity to pursue a attract talented young mathematicians to the subject. Full PhD in the West. Quite a few came to Penn State, which of energy and ideas, he immediately started a weekly sem- was already known for its extensive program in dynamics.
    [Show full text]
  • Scientific Workplace· • Mathematical Word Processing • LATEX Typesetting Scientific Word· • Computer Algebra
    Scientific WorkPlace· • Mathematical Word Processing • LATEX Typesetting Scientific Word· • Computer Algebra (-l +lr,:znt:,-1 + 2r) ,..,_' '"""""Ke~r~UrN- r o~ r PooiliorK 1.931'J1 Po6'lf ·1.:1l26!.1 Pod:iDnZ 3.881()2 UfW'IICI(JI)( -2.801~ ""'"""U!NecteoZ l!l!iS'11 v~ 0.7815399 Animated plots ln spherical coordln1tes > To make an anlm.ted plot In spherical coordinates 1. Type an expression In thr.. variables . 2 WMh the Insertion poilt In the expression, choose Plot 3D The next exampfe shows a sphere that grows ftom radius 1 to .. Plot 3D Animated + Spherical The Gold Standard for Mathematical Publishing Scientific WorkPlace and Scientific Word Version 5.5 make writing, sharing, and doing mathematics easier. You compose and edit your documents directly on the screen, without having to think in a programming language. A click of a button allows you to typeset your documents in LAT£X. You choose to print with or without LATEX typesetting, or publish on the web. Scientific WorkPlace and Scientific Word enable both professionals and support staff to produce stunning books and articles. Also, the integrated computer algebra system in Scientific WorkPlace enables you to solve and plot equations, animate 20 and 30 plots, rotate, move, and fly through 3D plots, create 3D implicit plots, and more. MuPAD' Pro MuPAD Pro is an integrated and open mathematical problem­ solving environment for symbolic and numeric computing. Visit our website for details. cK.ichan SOFTWARE , I NC. Visit our website for free trial versions of all our products. www.mackichan.com/notices • Email: info@mac kichan.com • Toll free: 877-724-9673 It@\ A I M S \W ELEGRONIC EDITORIAL BOARD http://www.math.psu.edu/era/ Managing Editors: This electronic-only journal publishes research announcements (up to about 10 Keith Burns journal pages) of significant advances in all branches of mathematics.
    [Show full text]
  • April/May 2009 | Volume 29 Number 3 MAA FOCUS the Newsmagazine
    MAA FOCUS The Newsmagazine of the Mathematical Association of America April/May 2009 | Volume 29 Number 3 WHAT’S INSIDE 9 ............Technology in Support of the Classroom 10 ............How to Excel at Math Transformation 19 ............Las Chicas’ View of Las Chicas de Matematicas 21 ............MathFest Portland, OR August 6–8, 2009 FOCUS_09_April_MayFINAL.indd 1 3/12/09 10:19:51 AM MAA FOCUS is published by the Mathematical Association of America in January, February/March, MAA FOCUS April/May, August/September, October/ November, and December/January. Editor: Fernando Gouvêa, Colby College Volume 29 | Issue 3 [email protected] Managing Editor: Carol Baxter, MAA [email protected] 3 Sylvia Bozeman Receives AAAS Mentor Award Senior Writer: Harry Waldman, MAA 3 Maria Gordina Wins 2009 Michler Prize [email protected] 4 Math Teachers’ Circles Connect Mathematicians with Please address advertising inquiries to: Middle School Teachers [email protected] Brian Conrey, Brianna Donaldson, and Tatiana Shubin David Bressoud President: 6 The Math Circle Summer Institute at Notre Dame First Vice President: Elizabeth Mayfield Bob and Ellen Kaplan Second Vice President: Daniel J. Teague 8 Teaching Time Savers: Secretary: Martha J. Siegel Student-Written Executive Summaries Associate Secretary: Gerard Venema Susan Martonosi Treasurer: John W. Kenelly 9 Technology in Support of the Classroom David M. Bressoud Executive Director: Tina H. Straley 10 How to Excel at Math Transformation Director of Publications for Journals and Communications: Ivars Peterson John Loase MAA FOCUS Editorial Board: Donald 12 Knowing What it Means to “Know Your Audience” J. Albers; Robert Bradley; Joseph Gallian; Aaron Luttman and Rachel Schwell Jacqueline Giles; Colm Mulcahy; Michael Orrison; Peter Renz; Sharon Cutler Ross; 14 MAA National Elections Coming Up in April and May 2009 Annie Selden; Hortensia Soto-Johnson; 16 What We Learned… Peter Stanek; Ravi Vakil.
    [Show full text]
  • Jürgen K. Moser 1928–1999
    Jürgen K. Moser 1928–1999 A Biographical Memoir by Paul H. Rabinowitz ©2015 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. J Ü RGEN KURT MOSER July 4, 1928–December 17, 1999 Elected to the NAS, 1971 After the death of Jürgen Moser, one of the world’s great mathematicians, the American Mathematical Society published a memorial article about his research. It is well worth beginning here with a lightly edited version of the brief introductory remarks I wrote then: One of those rare people with a gift for seeing mathematics as a whole, Moser was very much aware of its connections to other branches of science. His research had a profound effect on mathematics as well as on astronomy and physics. He made deep and important contributions to an extremely broad range of questions in dynamical systems and celestial mechanics, partial differen- By Paul H. Rabinowitz tial equations, nonlinear functional analysis, differ- ential and complex geometry, and the calculus of variations. To those who knew him, Moser exemplified both a creative scientist and a human being. His standards were high and his taste impeccable. His papers were elegantly written. Not merely focused on his own path- breaking research, he worked successfully for the well-being of math- ematics in many ways. He stimulated several generations of younger people by his penetrating insights into their problems, scientific and otherwise, and his warm and wise counsel, concern, and encouragement. My own experience as his student was typical: then and afterwards I was made to feel like a member of his family.
    [Show full text]
  • A Halmos-Von Neumann Theorem for Model Set Dynamical Systems
    Recent Progress in Dynamics MSRI Publications Volume 54, 2007 A Halmos–von Neumann theorem for model sets, and almost automorphic dynamical systems E. ARTHUR ROBINSON, JR. Dedicated to Anatole Katok in celebration of his 60th birthday. ABSTRACT. A subset ˙ ޒd is the spectrum of a model set ޒd if and only if it is a countable subgroup. The same result holds for ˙ G for a large class of locally compact abelian groups G. b 1. Introduction A model set is a special kind of uniformly discrete and relatively dense subset of a locally compact abelian group G. Model sets were first studied systematically in 1972 by Yves Meyer [15], who considered them in the context of Diophantine problems in harmonic analysis. More recently, model sets have played a prominent role in the theory of quasicrystals, beginning with N. G. de Bruijn’s 1981 discovery [4] that the vertices of a Penrose tiling are a model set. Much of the interest in model sets is due to the fact that although they are aperiodic, model sets have enough “almost periodicity” to give them a discrete Fourier transform. This corresponds to spots, or Bragg peaks, in the X-ray diffraction pattern of a quasicrystal. In this paper, we study model sets from the point of view of ergodic theory and topological dynamics. A translation invariant collection X of model sets in G has a natural topology, and with respect to this topology the translation action of G on X is continuous. We think of this action as a model set dynamical system.
    [Show full text]
  • New Members: Class of 2004
    New Members: Class of 2004 Class I: a role in establishing a mathe- sis of semi-simple groups and the Moses H. W. Chan matical theory for liquid crystals, spectral theory and cohomology Pennsylvania State University, Mathematical and for both the static and dynamic of their quotients. Work encom- University Park, PA Physical Sciences cases. Made contributions to the passes Lie theory, Riemannian Evan Pugh Professor of Physics. study of topological defects. Estab- geometry, differential equations, Conducted experimental studies Section 1: Mathematics lished the dynamical laws for automorphic forms, algebraic of phase transitions in quantum the Ginzburg-Landau vortices in geometry, mathematical physics, and classical fluids. Elucidated superconductivity and the topo- and quantum computing. Authored David Aldous the effects of reduced dimensions, logical solitons in superfluids. monographs on the cohomology restricted geometries, and impu- University of California, of discrete groups, representation rities on phase behavior. Dis- Berkeley, CA Yuri I. Manin theory, and invariant theory. covered the supersolid phase in Professor of Statistics. Leading Northwestern University, Evanston, helium. ½gure in discrete probability, the IL; Max Planck Institute for Mathe- Lai-Sang Young study of random processes on ½- matics, Bonn, Germany; Steklov New York University, Courant Paul A. Fleury nite structures. Made contribu- Mathematical Institute, Moscow, Institute of Mathematical Sciences, Yale University, New Haven, CT tions in several areas of probabil- Russia New York, NY Dean of Engineering and Freder- ity theory: rates of convergence to Trustee Chair and Professor of Professor of Mathematics. Expert equilibrium in ½nite-state Markov ick William Beinecke Professor Mathematics; Scienti½c Member; in the theory of dynamical sys- of Engineering and Applied Phys- chains, Poisson approximations Professor.
    [Show full text]
  • New Catalog Template
    New and Recent Books in Mathematics Contents Mail orders All orders from individuals must be prepaid or charged on American Express, VISA Highlights.............................................p. 3 or MasterCard. Forthcoming titles can be ordered now and will be shipped when they become available. Please feel free to share this information with your colleagues. Logic and Combinatorics......................p. 5 The order form may be photocopied. Algebra................................................p. 7 Phone orders Number Theory ....................................p. 9 To place a telephone order of $25 or more between the hours of 9:00 AM and 5:00 PM ET Monday through Friday: call 1-800-872-7423 in the U.S. and Canada. Geometry and Topology.....................p. 11 In Mexico call 55 5519 59 39. Mathematical Physics .........................p. 14 Library orders Analysis and Probability......................p. 16 Libraries are encouraged to use their wholesalers. Statistics and Finance .........................p. 20 Text Dynamical Systems.............................p. 22 Please visit us at www.cambridge.org/us/textbooks to request an examination copy of a textbook. You may also speak with a college sales representative toll free at Fluid Dynamics and Solid Mechanics...p. 23 1 866 257 3385 or email your request to [email protected]. Applied and Numerical Math..............p. 25 Book Proposals Computation......................................p. 27 We are always pleased to hear about new book proposals. Please contact our History and Philosophy of Mathematics editor, Roger Astley, at [email protected] Mathematics...................................p. 28 Web Site General Interest..................................p. 29 To view a complete listing of Cambridge University Press books in print, please visit our web site at: www.cambridge.org.
    [Show full text]
  • Geometry, Rigidity, and Group Actions (Chicago Lectures in Mathematics)
    Geometry, Rigidity, and Group Actions chicago lectures in mathematics series Editors: Spencer J. Bloch, Peter Constantin, Benson Farb, Norman R. Lebovitz, Carlos Kenig, and J. P. May Other Chicago Lectures in Mathematics titles available from the University of Chicago Press Simplical Objects in Algebraic Topology, by J. Peter May (1967, 1993) Fields and Rings, Second Edition, by Irving Kaplansky (1969, 1972) Lie Algebras and Locally Compact Groups, by Irving Kaplansky (1971) Several Complex Variables, by Raghavan Narasimhan (1971) Torsion-Free Modules, by Eben Matlis (1973) Stable Homotopy and Generalised Homology, by J. F. Adams (1974) Rings with Involution, by I. N. Herstein (1976) Theory of Unitary Group Representation, by George V. Mackey (1976) Infinite-Dimensional Optimization and Convexity, by Ivar Ekeland and Thomas Turnbull (1983) Commutative Semigroup Rings, by Robert Gilmer (1984) Navier-Stokes Equations, by Peter Constantin and Ciprian Foias (1988) Essential Results of Functional Analysis, by Robert J. Zimmer (1990) Fuchsian Groups, by Svetlana Katok (1992) Topological Classification of Stratified Spaces, by Shmuel Weinberger (1994) Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture, by Lionel Schwartz (1994) Geometry of Nonpositively Curved Manifolds, by Patrick B. Eberlein (1996) Lectures on Exceptional Lie Groups, by J. F. Adams (1996) Dimension Theory in Dynamical Systems: Contemporary Views and Applications, by Yakov B. Pesin (1997) A Concise Course in Algebraic Topology, by J. P. May (1999)
    [Show full text]
  • Reviewed by Thomas Ward
    Citations From References: 1 Previous Up Next Article From Reviews: 0 MR2342699 (2008i:37001) 37-02 (01A60 28D20 37-03 37A35) Katok, Anatole (1-PAS) Fifty years of entropy in dynamics: 1958–2007. J. Mod. Dyn. 1 (2007), no. 4, 545–596. This is an attractive overview of the role played by “entropy” in the study of dynamical systems since its discovery by Kolmogorov at the end of the 1950s. Several features make it a particularly interesting read. First, the author has been an active participant in the field, often very close to the most exciting and influential developments. Second, the article does not attempt to provide a (potentially indigestible) complete history shaped by time, but instead an overview exposing the mathematical story, taking full advantage of later insights and developments. Third, it is written with candour (for example, one of the many interesting footnotes records the extent to which the disjointness paper of Furstenberg in 1967 was underestimated), with a vision of great extent in time and breadth in topic, with much detailed knowledge (for example, the ergodic-theory aspect of Wiener’s monograph of 1958 was new to me), and with great authority (both mathematical and personal: the author interacted directly with many of the key contributors both in the “Moscow school” and later in the United States). The breadth of the pervasive influence of entropy is re- flected in the main sections: Early history of entropy; Isomorphism problem; Topological entropy, variational principle, symbolic dynamics; Entropy and nonuniform hyperbolicity; Entropy as a source of rigidity; Rigidity of positive entropy measures for higher-rank abelian actions.
    [Show full text]
  • Invariant Measures and the Set of Exceptions to Littlewood's Conjecture
    INVARIANT MEASURES AND THE SET OF EXCEPTIONS TO LITTLEWOOD’S CONJECTURE MANFRED EINSIEDLER, ANATOLE KATOK, AND ELON LINDENSTRAUSS Abstract. We classify the measures on SL(k, R)/ SL(k, Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood’s conjecture has Hausdorff dimension zero. 1. Introduction 1.1. Number theory and dynamics. There is a long and rich tradition of applying dynamical methods to number theory. In many of these applica- tions, a key role is played by the space SL(k, R)/ SL(k, Z) which can be iden- k tified as the space of unimodular lattices in R . Any subgroup H < SL(k, R) acts on this space in a natural way, and the dynamical properties of such actions often have deep number theoretical implications. A significant landmark in this direction is the solution by G.A. Margulis [23] of the long-standing Oppenheim Conjecture through the study of the action of a certain subgroup H on the space of unimodular lattices in three space. This conjecture, posed by A. Oppenheim in 1929, deals with den- sity properties of the values of indefinite quadratic forms in three or more variables. So far there is no proof known of this result in its entirety which avoids the use of dynamics of homogeneous actions. An important property of the acting group H in the case of the Oppen- heim Conjecture is that it is generated by unipotents: i.e.
    [Show full text]