Ricci Flow and the Sphere Theorem
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Marcel Berger Remembered
Marcel Berger Remembered Claude LeBrun, Editor and Translator Gérard Besson EDITOR’S NOTE. Claude LeBrun has kindly assembled this memorial for Marcel Berger. This month also Marcel Berger,1 one of the world’s leading differential marks the conference “Riemannian Geometry Past, geometers and a corresponding member of the French Present and Future: An Homage to Marcel Berger” at Academy of Sciences for half a century, passed away on IHÉS. October 15, 2016, at the age of eighty-nine. Marcel Berger’s contributions to geometry were both broad and deep. The classification of Riemannian holo- nomy groups provided by his thesis has had a lasting impact on areas ranging from theoretical physics to alge- braic geometry. His 1960 proof that a complete oriented even-dimensional manifold with strictly quarter-pinched positive curvature must be a topological sphere is the Claude LeBrun is professor of mathematics at Stony Brook Univer- sity. His email address is [email protected]. Gérard Besson is CNRS Research Director at Institut Fourier, Uni- For permission to reprint this article, please contact: versité Grenoble, France. His email address is g.besson@ujf [email protected]. -grenoble.fr. DOI: http://dx.doi.org/10.1090/noti1605 1Pronounced bare-ZHAY. December 2017 Notices of the AMS 1285 research director by the CNRS. He served as president of the French Mathematical Society (SMF) during the period of 1979–80 and in this capacity helped oversee the foundation of the International Center for Mathematical Workshops (CIRM) in Luminy. In 1985 he then became director of the Institute for Higher Scientific Studies (IHÉS) in Bures-sur-Yvette, a position in which he served until 1993, when he was succeeded by his former student Jean-Pierre Bourguignon. -
The Geometry of Asymptotically Hyperbolic Manifolds a Dissertation Submitted to the Department of Mathematics and the Committee
THE GEOMETRY OF ASYMPTOTICALLY HYPERBOLIC MANIFOLDS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Otis Chodosh June 2015 © 2015 by Otis Avram Chodosh. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/mp634xn8004 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Simon Brendle, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Michael Eichmair, Co-Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Leon Simon I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Brian White Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. -
All That Math Portraits of Mathematicians As Young Researchers
Downloaded from orbit.dtu.dk on: Oct 06, 2021 All that Math Portraits of mathematicians as young researchers Hansen, Vagn Lundsgaard Published in: EMS Newsletter Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hansen, V. L. (2012). All that Math: Portraits of mathematicians as young researchers. EMS Newsletter, (85), 61-62. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. -
Hamilton's Ricci Flow
The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland. -
EMS Newsletter September 2012 1 EMS Agenda EMS Executive Committee EMS Agenda
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. Matemática, Faculdade Facultad de Matematicas de Ciências, Edifício C6, Universidad Complutense Piso 2 Campo Grande Mathematical de Madrid 1749-006 Lisboa, Portugal e-mail: [email protected] Plaza de Ciencias 3, 28040 Madrid, Spain Eva-Maria Feichtner e-mail: [email protected] (2012–2015) Society Department of Mathematics Lucia Di Vizio (2012–2016) Université de Versailles- University of Bremen St Quentin 28359 Bremen, Germany e-mail: [email protected] Laboratoire de Mathématiques Newsletter No. 85, September 2012 45 avenue des États-Unis Eva Miranda (2010–2013) 78035 Versailles cedex, France Departament de Matemàtica e-mail: [email protected] Aplicada I EMS Agenda .......................................................................................................................................................... 2 EPSEB, Edifici P Editorial – S. Jackowski ........................................................................................................................... 3 Associate Editors Universitat Politècnica de Catalunya Opening Ceremony of the 6ECM – M. -
Book Reviews
BOOK REVIEWS BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 1, July 1984 ©1984 American Mathematical Society 0273-0979/84 $1.00 + $.25 per page Riemannian geometry, by Wilhelm Klingenberg, de Gruyter Studies in Mathematics, Vol. 1, Walter de Gruyter & Co., Berlin, 1982, x + 396 pp., $49.00. ISBN 3-1100-8373-5 The basic concepts of Riemannian geometry have become useful in a surprising variety of mathematical subjects. The terminology of manifolds, bundles, Riemannian metrics, and connections has become a lingua franca over much of partial differential equations, mathematical physics, and algebraic geometry, among other fields. This increasingly widespread use of the terminology and methods of geometry has tended to obscure the fact that Riemannian geometry as such is a subject with a quite precisely focused program, namely, to determine how the topology of a manifold is influenced by the local properties of its metric structure. These local properties are usually formulated in terms of curvature; this formulation is justified by the theorem of E. Cartan that the curvature tensor and its covariant derivatives of all orders at a point determine the formal Taylor expansion of the metric at that point. The program of obtaining global topological information from local metric information actually applies only to complete Riemannian manifolds, i.e., those that are complete as metric spaces when the distance between points is defined to be the infimum of the Riemannian-metric arc length of curves joining the two points. This restriction to complete manifolds, long a standard one in the subject, has been given explicit justification by a result of M. -
Mathematics People
NEWS Mathematics People Tardos Named Goldfarb and Nocedal Kovalevsky Lecturer Awarded 2017 von Neumann Éva Tardos of Cornell University Theory Prize has been chosen as the 2018 AWM- SIAM Sonia Kovalevsky Lecturer by Donald Goldfarb of Columbia the Association for Women in Math- University and Jorge Nocedal of ematics (AWM) and the Society for Northwestern University have been Industrial and Applied Mathematics awarded the 2017 John von Neu- (SIAM). She was honored for her “dis- mann Theory Prize by the Institute tinguished scientific contributions for Operations Research and the to the efficient methods for com- Management Sciences (INFORMS). binatorial optimization problems According to the prize citation, “The Éva Tardos on graphs and networks, and her award recognizes the seminal con- work on issues at the interface of tributions that Donald Goldfarb computing and economics.” According to the prize cita- Jorge Nocedal and Jorge Nocedal have made to the tion, she is considered “one of the leaders in defining theory and applications of nonlinear the area of algorithmic game theory, in which algorithms optimization over the past several decades. These contri- are designed in the presence of self-interested agents butions cover a full range of topics, going from modeling, governed by incentives and economic constraints.” With to mathematical analysis, to breakthroughs in scientific Tim Roughgarden, she was awarded the 2012 Gödel Prize computing. Their work on the variable metric methods of the Association for Computing Machinery (ACM) for a paper “that shaped the field of algorithmic game theory.” (BFGS and L-BFGS, respectively) has been extremely in- She received her PhD in 1984 from Eötvös Loránd Univer- fluential.” sity. -
Curvature Cones and the Ricci Flow
Institut Fourier — Université de Grenoble I Actes du séminaire de Théorie spectrale et géométrie Thomas RICHARD Curvature cones and the Ricci flow. Volume 31 (2012-2014), p. 197-220. <http://tsg.cedram.org/item?id=TSG_2012-2014__31__197_0> © Institut Fourier, 2012-2014, tous droits réservés. L’accès aux articles du Séminaire de théorie spectrale et géométrie (http://tsg.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://tsg.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Séminaire de théorie spectrale et géométrie Grenoble Volume 31 (2012-2014) 197-220 CURVATURE CONES AND THE RICCI FLOW. Thomas Richard Abstract. — This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of preserved curvature conditions and how they have been used to derive geometric results, in particular sphere theorems. We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters. The Ricci flow is the following evolution equation: ( ∂g = −2 Ric (0.1) ∂t g(t) g(t) = g0 where (g(t))t∈[0,T ) is a one-parameter family of smooth Riemannian metrics on a fixed manifold M, and g0 is a given smooth Riemannian metric on M. It was introduced by R. Hamilton in 1982 ([11]), where it was used to study the topology of compact 3-manifolds with positive Ricci curvature. -
Examples of Riemannian Manifolds with Non-Negative Sectional Curvature
EXAMPLES OF RIEMANNIAN MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE WOLFGANG ZILLER Manifolds with non-negative sectional curvature have been of interest since the beginning of global Riemannian geometry, as illustrated by the theorems of Bonnet-Myers, Synge, and the sphere theorem. Some of the oldest conjectures in global Riemannian geometry, as for example the Hopf conjecture on S2 £ S2, also ¯t into this subject. For non-negatively curved manifolds, there are a number of obstruction theorems known, see Section 1 below and the survey by Burkhard Wilking in this volume. It is somewhat surprising that the only further obstructions to positive curvature are given by the classical Bonnet-Myers and Synge theorems on the fundamental group. Although there are many examples with non-negative curvature, they all come from two basic constructions, apart from taking products. One is taking an isometric quotient of a compact Lie group equipped with a biinvariant metric and another a gluing procedure due to Cheeger and recently signi¯cantly generalized by Grove-Ziller. The latter examples include a rich class of manifolds, and give rise to non-negative curvature on many exotic 7-spheres. On the other hand, known manifolds with positive sectional curvature are very rare, and are all given by quotients of compact Lie groups, and, apart from the classical rank one symmetric spaces, only exist in dimension below 25. Due to this lack of knowledge, it is therefore of importance to discuss and understand known examples and ¯nd new ones. In this survey we will concentrate on the description of known examples, although the last section also contains suggestions where to look for new ones. -
Arxiv:1710.03952V3 [Math.MG] 11 Jul 2021 Example Lp-Unit Spheres)
THE ISOPERIMETRIC INEQUALITY ON COMPACT RANK ONE SYMMETRIC SPACES AND BEYOND YASHAR MEMARIAN Abstract. Klartag's needle decomposition technique enables one to obtain strong isoperimetric inequalities on Riemannian manifolds other than the clas- sical known examples. As a result, in this paper, we obtain sharp isoperimetric inequalities for compact rank one symmetric spaces (CROSS). Namely, for the n real projective space RP , we demonstrate that the isoperimetric regions are k n given by either the geodesic balls or tubes around some RP ⊂ RP . For the n complex projective space CP , the isoperimetric regions are given by either k n the geodesic balls or tubes around some CP ⊂ CP . And for the quaternionic projective space, the isoperimetric regions are given by either the geodesic balls k n or tubes around some HP ⊂ HP . 1. Introduction Isoperimetric problems are some of the oldest problems in geometry. Given a space, one looks for domains of a given volume with the least boundary area. It is known that in model spaces (i.e. Euclidean spaces, spheres and hyperbolic spaces), for every given number v, the intrinsic balls with volume v have the least surface area among every domain with the same volume. When we leave the world of model spaces, or when we are dealing with geometric spaces with boundary, the solution to isoperimetric problems has provided real difficulties. Of course, when the number v is small enough, for spaces which locally look like Euclidean spaces (for instance manifolds), one expects that the solution of isoperimetric problems still would be the metric balls. However, for larger v and shapes with non-constant curvature or with non-smooth boundaries, the isoperimetric problem in general is very hard to solve. -
DMS COV Report
Report of the 2010 Committee of Visitors Division of Mathematical Sciences National Science Foundation 26-28 April 2010 Submitted on behalf of the Committee by C. David Levermore, Chair to H. Edward Seidel, Acting Assistant Director Mathematical and Physical Sciences 1. Charge, Organization, and Procedures 2. Major Findings, Recommendations, and Concerns 3. A More Detailed Look at Disciplinary Programs 4. Responses to Template Questions 5. Responses to Additional Questions Appendix A. Committee Membership Appendix B. Subcommittee Structure Appendix C. Conflict of Interest Report Appendix D. Meeting Agenda 1 1. Charge, Organization, and Procedures The 2010 Committee of Visitors (COV) for the Division of Mathematical Sciences (DMS) of the National Science Foundation (NSF) was charged by Edward Seidel, Acting Assistant Director for Mathematical and Physical Sciences (MPS), to address and report on: • the integrity and efficacy of processes used to solicit, review, recommend, and document proposal actions; • the quality and significance of the results of the Division's programmatic investments; • the relationship between award decisions, program goals, and Foundation-wide programs and strategic goals; • the Division's balance, priorities, and future directions; • the Division's response to the prior COV report of 2007; and • any other issues that the COV feels are relevant to the review. Following NSF guidelines, the COV consisted of a diverse group of members representing different work environments (research-intensive public and private universities, primarily undergraduate colleges and universities, private industry, other Federal government agencies or laboratories, and non-U.S. institutions), gender, ethnicity, and geographical location. A list of the COV members and their affiliations is given in Appendix A. -
Comparison Theorems in Riemannian Geometry
COMPARISON THEOREMS IN RIEMANNIAN GEOMETRY JEFF CHEEGER DAVID G. EBIN AMS CHELSEA PUBLISHING !MERICAN -ATHEMATICAL 3OCIETY s 0ROVIDENCE 2HODE )SLAND COMPARISON THEOREMS IN RIEMANNIAN GEOMETRY David G. Ebin and Jeff Cheeger G.EbinandJeff David Photo courtesy of Ann Billingsley COMPARISON THEOREMS IN RIEMANNIAN GEOMETRY JEFF CHEEGER DAVID G. EBIN AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ N ΕΙΣΙΤΩ S A O C C I I R E E T ΑΓΕΩΜΕ Y M A F O 8 U 88 NDED 1 2000 Mathematics Subject Classification. Primary 53C20; Secondary 58E10. For additional information and updates on this book, visit www.ams.org/bookpages/chel-365 Library of Congress Cataloging-in-Publication Data Cheeger, Jeff. Comparison theorems in riemannian geometry / Jeff Cheeger, David G. Ebin. p. cm. — (AMS Chelsea Publishing) Originally published: Amsterdam : North-Holland Pub. Co. ; New York : American Elsevier Pub. Co., 1975, in series: North-Holland mathematical library ; v. 9. Includes bibliographical references and index. ISBN 978-0-8218-4417-5 (alk. paper) 1. Geometry, Riemannian. 2. Riemannian manifolds. I. Ebin, D. G. II. American Mathe- matical Society. III. Title. QA649 .C47 2008 516.373—dc22 2007052113 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.