1999 Newsletter; UC Davis Departmet of Mathematics
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Prospects in Topology
Annals of Mathematics Studies Number 138 Prospects in Topology PROCEEDINGS OF A CONFERENCE IN HONOR OF WILLIAM BROWDER edited by Frank Quinn PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1995 Copyright © 1995 by Princeton University Press ALL RIGHTS RESERVED The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton Academic Press 10 987654321 Library of Congress Cataloging-in-Publication Data Prospects in topology : proceedings of a conference in honor of W illiam Browder / Edited by Frank Quinn. p. cm. — (Annals of mathematics studies ; no. 138) Conference held Mar. 1994, at Princeton University. Includes bibliographical references. ISB N 0-691-02729-3 (alk. paper). — ISBN 0-691-02728-5 (pbk. : alk. paper) 1. Topology— Congresses. I. Browder, William. II. Quinn, F. (Frank), 1946- . III. Series. QA611.A1P76 1996 514— dc20 95-25751 The publisher would like to acknowledge the editor of this volume for providing the camera-ready copy from which this book was printed PROSPECTS IN TOPOLOGY F r a n k Q u in n , E d it o r Proceedings of a conference in honor of William Browder Princeton, March 1994 Contents Foreword..........................................................................................................vii Program of the conference ................................................................................ix Mathematical descendants of William Browder...............................................xi A. Adem and R. J. Milgram, The mod 2 cohomology rings of rank 3 simple groups are Cohen-Macaulay........................................................................3 A. -
Problems in Low-Dimensional Topology
Problems in Low-Dimensional Topology Edited by Rob Kirby Berkeley - 22 Dec 95 Contents 1 Knot Theory 7 2 Surfaces 85 3 3-Manifolds 97 4 4-Manifolds 179 5 Miscellany 259 Index of Conjectures 282 Index 284 Old Problem Lists 294 Bibliography 301 1 2 CONTENTS Introduction In April, 1977 when my first problem list [38,Kirby,1978] was finished, a good topologist could reasonably hope to understand the main topics in all of low dimensional topology. But at that time Bill Thurston was already starting to greatly influence the study of 2- and 3-manifolds through the introduction of geometry, especially hyperbolic. Four years later in September, 1981, Mike Freedman turned a subject, topological 4-manifolds, in which we expected no progress for years, into a subject in which it seemed we knew everything. A few months later in spring 1982, Simon Donaldson brought gauge theory to 4-manifolds with the first of a remarkable string of theorems showing that smooth 4-manifolds which might not exist or might not be diffeomorphic, in fact, didn’t and weren’t. Exotic R4’s, the strangest of smooth manifolds, followed. And then in late spring 1984, Vaughan Jones brought us the Jones polynomial and later Witten a host of other topological quantum field theories (TQFT’s). Physics has had for at least two decades a remarkable record for guiding mathematicians to remarkable mathematics (Seiberg–Witten gauge theory, new in October, 1994, is the latest example). Lest one think that progress was only made using non-topological techniques, note that Freedman’s work, and other results like knot complements determining knots (Gordon- Luecke) or the Seifert fibered space conjecture (Mess, Scott, Gabai, Casson & Jungreis) were all or mostly classical topology. -
The Shape of Inner Space Provides a Vibrant Tour Through the Strange and Wondrous Possibility SPACE INNER
SCIENCE/MATHEMATICS SHING-TUNG $30.00 US / $36.00 CAN Praise for YAU & and the STEVE NADIS STRING THEORY THE SHAPE OF tring theory—meant to reconcile the INNER SPACE incompatibility of our two most successful GEOMETRY of the UNIVERSE’S theories of physics, general relativity and “The Shape of Inner Space provides a vibrant tour through the strange and wondrous possibility INNER SPACE THE quantum mechanics—holds that the particles that the three spatial dimensions we see may not be the only ones that exist. Told by one of the Sand forces of nature are the result of the vibrations of tiny masters of the subject, the book gives an in-depth account of one of the most exciting HIDDEN DIMENSIONS “strings,” and that we live in a universe of ten dimensions, and controversial developments in modern theoretical physics.” —BRIAN GREENE, Professor of © Susan Towne Gilbert © Susan Towne four of which we can experience, and six that are curled up Mathematics & Physics, Columbia University, SHAPE in elaborate, twisted shapes called Calabi-Yau manifolds. Shing-Tung Yau author of The Fabric of the Cosmos and The Elegant Universe has been a professor of mathematics at Harvard since These spaces are so minuscule we’ll probably never see 1987 and is the current department chair. Yau is the winner “Einstein’s vision of physical laws emerging from the shape of space has been expanded by the higher them directly; nevertheless, the geometry of this secret dimensions of string theory. This vision has transformed not only modern physics, but also modern of the Fields Medal, the National Medal of Science, the realm may hold the key to the most important physical mathematics. -
Dynamics of D-Branes II. the Standard Action---An Analogue of The
February 2017 arXiv:yymm.nnnnn [hep-th] D(13.3): standard Dynamics of D-branes II. The standard action | an analogue of the Polyakov action for (fundamental, stacked) D-branes Chien-Hao Liu and Shing-Tung Yau Abstract (ρ,h;Φ;g;B;C) We introduce a new action Sstandard for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This `standard action' is abstractly a non-Abelian gauged sigma model | based on maps ' :(XAz;E; r) ! Y from an Azumaya/matrix manifold XAz with a fundamental module E with a connection r to Y | enhanced by the dila- ton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples ('; r) to Ramond-Ramond field. In a special situation, this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. With the analysis developed in D(13.1) (arXiv:1606.08529 [hep-th]) for such maps and an improved understanding of the hierarchy of various admissible conditions on the pairs ('; r) beyond D(13.2.1) (arXiv:1611.09439 [hep-th]) and how they resolve the built-in obstruction to pull- push of covariant tensors under a map from a noncommutative manifold to a commutative manifold, we develop further in this note some covariant differential calculus needed and apply them to work out the first variation | and hence the corresponding equations of mo- tion for D-branes | of the standard action and the second variation of the kinetic term for maps and the dilaton term in this action. -
Introduction to 3-Manifolds
Introduction to 3-Manifolds Jennifer Schultens Graduate Studies in Mathematics Volume 151 American Mathematical Society https://doi.org/10.1090//gsm/151 Introduction to 3-Manifolds Jennifer Schultens Graduate Studies in Mathematics Volume 151 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 57N05, 57N10, 57N16, 57N40, 57N50, 57N75, 57Q15, 57Q25, 57Q40, 57Q45. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-151 Library of Congress Cataloging-in-Publication Data Schultens, Jennifer, 1965– Introduction to 3-manifolds / Jennifer Schultens. pages cm — (Graduate studies in mathematics ; v. 151) Includes bibliographical references and index. ISBN 978-1-4704-1020-9 (alk. paper) 1. Topological manifolds. 2. Manifolds (Mathematics) I. Title. II. Title: Introduction to three-manifolds. QA613.2.S35 2014 514.34—dc23 2013046541 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the author. -
The Work of John Milnor
THE WORK OF JOHN MILNOR W.T. GOWERS 1. The hardest IQ question ever What is the next term in the following sequence: 1, 2, 5, 14, 41, 122? One can imagine such a question appearing on an IQ test. And one doesn't have to stare at it for too long to see that each term is obtained by multiplying the previous term by 3 and subtracting 1. Therefore, the next term is 365. If you managed that, then you might find the following question more challenging. What is the next term in the sequence 1; 1; 28; 2; 8; 6; 992; 1; 3; 2; 16256; 2; 16; 16; 523264? I will leave that question hanging for now, but I promise to reveal the answer later. 2. A giant of modern mathematics There are many mathematicians with extraordinary achievements to their names, and many who have won major prizes. But even in this illustrious company, John Milnor stands out. It is not just that he has proved several famous theorems: it is also that the areas in which he has made fundamental contributions have been very varied, and that he is renowned as a quite exceptionally gifted expositor. As a result, his influence can be felt all over modern mathematics. I will not be able to do more than scratch the surface of what he has done, partly because of the limited time I have and partly because I work in a different field. The areas that he has worked in include differential topology, K-theory, group theory, game theory, and dynamical systems. -
Virtual Properties of 3-Manifolds Dedicated to the Memory of Bill Thurston
Virtual properties of 3-manifolds dedicated to the memory of Bill Thurston Ian Agol ∗ Abstract. We will discuss the proof of Waldhausen's conjecture that compact aspheri- cal 3-manifolds are virtually Haken, as well as Thurston's conjecture that hyperbolic 3- manifolds are virtually fibered. The proofs depend on major developments in 3-manifold topology of the past decades, including Perelman's resolution of the geometrization con- jecture, results of Kahn and Markovic on the existence of immersed surfaces in hyperbolic 3-manifolds, and Gabai's sutured manifold theory. In fact, we prove a more general the- orem in geometric group theory concerning hyperbolic groups acting on CAT(0) cube complexes, concepts introduced by Gromov. We resolve a conjecture of Dani Wise about these groups, making use of the theory that Wise developed with collaborators including Bergeron, Haglund, Hsu, and Sageev as well as the theory of relatively hyperbolic Dehn filling developed by Groves-Manning and Osin. Mathematics Subject Classification (2010). Primary 57M Keywords. hyperbolic, 3-manifold 1. Introduction In Thurston's 1982 Bulletin of the AMS paper Three Dimensional Manifolds, Kleinian groups, and hyperbolic geometry [118], he asked 24 questions which have guided the last 30 years of research in the field. Four of the questions have to do with \virtual" properties of 3-manifolds: • Question 15 (paraphrased): Are Kleinian groups LERF? [76, Problem 3.76 (Hass)] • Question 16: \Does every aspherical 3-manifold have a finite-sheeted cover which is Haken?" This -
The Hauptvermutung Book
The Hauptvermutung Book A collection of papers on the topology of manifolds by A.A.Ranicki (editor), A.J.Casson, D.P.Sullivan, M.A.Armstrong, C.P.Rourke, and G.E.Cooke M. A. Armstrong Mathematics Department, University of Durham, Durham DH1 3LE, England, UK A. J. Casson Mathematics Department, University of California, Berkeley, CA 94720, USA G. E. Cookey A. A. Ranicki Department of Mathematics and Statistics, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK C. P. Rourke Mathematics Institute, University of Warwick, Coventry CV4 7AL, England, UK D. P. Sullivan Mathematics Department, City University of New York, Graduate Center, 33 West 42 Street, New York, NY10036, USA Contents Preface . 1 On the Hauptvermutung (by A.A.Ranicki, 1996) 1. Introduction . 3 2. The Polyhedral Hauptvermutung . 6 3. The Rochlin Invariant . 11 4. The Manifold Hauptvermutung . 13 5. Homology Manifolds . 23 References . 29 Generalisations and Applications of Block Bundles (by A.J.Casson, 1967) Introduction . 33 I. Block Bundles . 35 II. Homotopy Properties of Block Bundles . 43 III. Tangential Properties of Block Bundles . 49 IV. Periodicity of G=P L ....................55 V. Topologically Trivial Block Bundles . 59 References . 67 Triangulating Homotopy Equivalences and Homeomorphisms. Geometric Topology Seminar Notes (by D.P.Sullivan, 1967) Introduction . 69 I. Triangulating and Smoothing Homotopy Equivalences . 69 II. The Characteristic Bundle of a Homotopy Equivalence . 80 III. The Hauptvermutung . 93 References . 103 The Princeton Notes on the Hauptvermutung (by M.A.Armstrong, C.P.Rourke, G.E.Cooke) Preface (1996) . 105 I. The Hauptvermutung According to Lashof and Rothenberg (by M.A.Armstrong, 1968) 1. -
Fifty Years Ago: Topology in the 50'S and 60'S
Fifty Years Ago: Topology of Manifolds in the 50's and 60's John Milnor ¤ Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY. 11794-3660. e-mail: [email protected] June 27, 2006 The 1950's and 1960's were exciting times to study the topology of manifolds. This lecture will try to describe some of the more interesting developments. The ¯rst two sections describe work in dimension 3, and in dimensions n ¸ 5, while x3 discusses why it is often easier to work in higher dimensions. The last section is a response to questions from the audience. 1 3-Dimensional Manifolds. A number of mathematicians worked on 3-dimensional manifolds in the 50's. (I was certainly one of them.) But I believe that the most important contribu- tion was made by just one person. Christos Papakyriakopoulos had no regular academic position, and worked very much by himself, concentrating on old and di±cult problems. We were both in Princeton during this period, and I saw him fairly often, but had no idea that he was doing such important work. (In fact, I don't really remember talking to him|perhaps we were both too shy.) Let me try to explain what he accomplished. In 1910, Max Dehn had claimed a proof of the following lemma: If a piecewise linear map from a 2-simplex ¢ into a triangulated 3- manifold is one-to-one near @¢, and if the image of the interior is disjoint from the image of the boundary, then there exists a piecewise linear embedding of ¢ which agrees with the original map near @¢: As an easy corollary, he concluded that if the fundamental group of a knot com- plement is free cyclic, then there exists a spanning disk, so that the curve must be unknotted. -
Frederick W. Gehring : a Biographical Memoir
Frederick W. Gehring 1925–2012 A Biographical Memoir by Gaven J. Martin ©2017 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. FREDERICK WILLIAM GEHRING August 7, 1925–May 29, 2012 Elected to the NAS, 1989 Frederick W. Gehring was a hugely influential mathema- tician who spent most of his career at the University of Michigan (UM). Appointed to the faculty in 1955, he was the T. H. Hildebrandt Distinguished University Professor from 1987 on. Gehring’s major research contributions were to geometric function theory, particularly in higher dimensions n , where n ≥ 3. Gehring developed this field in close coordination with colleagues, primarily in Finland, over the three decades from 1960 to 1990. His seminal work in geometric function theory—notably, by making important connections with geometry and nonlinear Paul Bishop. G. Photography by permission of partial differential equations to solve major problems— helped drive the field forward. Over the course of his career, Gehring received numerous By Gaven J. Martin honors from the international mathematical community. He was invited three times to address the International Congress of Mathematicians—in Moscow in 1966, Vancouver in 1974, and Berkeley (a plenary lecture) in 1986—and was awarded honorary degrees from Cambridge University (1976), from Finland’s University of Helsinki (1979) and University of Jyväskylä (1990), and from the Norwegian University of Science and Technology (1997). In 1989, Gehring was elected to the American Academy of Arts and Sciences and the U.S. -
Some Applications of Ricci Flow to 3-Manifolds
Some applications of Ricci flow to 3-manifolds Sylvain Maillot November 6, 2018 1 Introduction The purpose of this text is to describe some applications of Ricci flow to ques- tions about the topology and geometry of 3-manifolds. The most spectacular achievement in the recent years was the proof of W. Thurston’s Geometriza- tion Conjecture by G. Perelman [Per02, Per03b, Per03a]. In Sections 2–6 we outline a proof of the Geometrization Conjecture which is based in part on Perelman’s ideas. This is joint work with L. Bessi`eres, G. Besson, M. Boileau, and J. Porti. Some details have already appeared in the preprint [BBB+07]; the rest will be contained in our forthcoming monograph [BBB+]. In Sec- tion 7 we announce some results on the topological classification of (possibly noncompact) 3-manifolds with positive scalar curvature. These results follow from an extension of those discussed in Section 6, and are joint work with L. Bessi`eres and G. Besson. Details will appear elsewhere. The theory of low-dimensional Ricci flows relies on techniques and in- sights from riemannian geometry, geometric analysis, and topology. In this text, we deliberately adopt the topologist’s viewpoint; thus we shall focus on the topological and geometric arguments rather than the analytic aspects. The Ricci flow is considered here as a tool to prove topological and geomet- ric theorems. Therefore, we sometimes do not prove the strongest possible estimates on Ricci flow solutions, but rather strengthen the theorems that arXiv:0801.4881v1 [math.DG] 31 Jan 2008 are used to deduce topological consequences (e.g. -
RESTON for MATH======:::---"", New and Recent Titles in Mathematics
Louisville Meetings (January 25-28)- Page 23 Notices of the American Mathematical Society january 1984, Issue 231 Volume 31, Number 1, Pages 1-136 Providence, Rhode Island USA ISSN 0002-9920 Calendar of AMS Meetings THIS CALENDAR lists all meetings which have been approved by the Council prior to the date this issue of the Notices was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the Ameri· can Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and second announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meet· ing. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the office of the Society in Providence. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline given below for the meeting. Note that the deadline for ab· stracts submitted for consideration for presentation at sp~cial sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcement and the list of organizers of special sessions.