Geometric Periodicity and the Invariants of Manifolds
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Ravi's Speech at the Banquet
Speech at the Conference on Conformal Geometry and Riemann Surfaces October 27, 2013 Ravi S. Kulkarni October 29, 2013 1 Greetings I am very happy today. I did not know that so many people loved me enough to gather at Queens College to wish me a healthy, long, and productive life over and above the 71 years I have already lived. It includes my teacher Shlomo Sternberg, present here on skype, and my \almost"-teachers Hyman Bass, and Cliff Earle. Alex Lubotzky came from Israel, Ulrich Pinkall from Germany, and Shiga from Japan. If I have counted correctly there are 14 people among the speakers who are above 65, and 5 below 65, of which only 3 in their 30s to 50s. There are many more in the audience who are in their 50s and below. I interpret this as: we old people have done something right. And of course that something right, is that we have done mathematics. The conference of this type is new for the Math department at Queens College, although it had many distinguished mathematicians like Arthur Sard, Leo Zippin, Banesh Hoffman, Edwin Moise, ... before, on its faculty. I find this Conference especially gratifying since I already went back to In- dia in 2001, enjoyed several leaves without pay, and finally retired from Queens College, in Feb 2008. However I keep coming back to Queens college and Grad- uate Center twice a year and enjoy my emeritus positions with all the office and library/computer advantages. For a long time, I felt that people here thought that I was an Indian in America. -
On the Stable Classification of Certain 4-Manifolds
BULL. AUSTRAL. MATH. SOC. 57N65, 57R67, 57Q10, 57Q20 VOL. 52 (1995) [385-398] ON THE STABLE CLASSIFICATION OF CERTAIN 4-MANIFOLDS ALBERTO CAVICCHIOLI, FRIEDRICH HEGENBARTH AND DUSAN REPOVS We study the s-cobordism type of closed orientable (smooth or PL) 4—manifolds with free or surface fundamental groups. We prove stable classification theorems for these classes of manifolds by using surgery theory. 1. INTRODUCTION In this paper we shall study closed connected (smooth or PL) 4—manifolds with special fundamental groups as free products or surface groups. For convenience, all manifolds considered will be assumed to be orientable although our results work also in the general case, provided the first Stiefel-Whitney classes coincide. The starting point for classifying manifolds is the determination of their homotopy type. For 4- manifolds having finite fundamental groups with periodic homology of period four, this was done in [12] (see also [1] and [2]). The case of a cyclic fundamental group of prime order was first treated in [23]. The homotopy type of 4-manifolds with free or surface fundamental groups was completely classified in [6] and [7] respectively. In particular, closed 4-manifolds M with a free fundamental group IIi(M) = *PZ (free product of p factors Z ) are classified, up to homotopy, by the isomorphism class of their intersection pairings AM : B2{M\K) x H2(M;A) —> A over the integral group ring A = Z[IIi(M)]. For Ui(M) = Z, we observe that the arguments developed in [11] classify these 4-manifolds, up to TOP homeomorphism, in terms of their intersection forms over Z. -
Math Spans All Dimensions
March 2000 THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA Math Spans All Dimensions April 2000 is Math Awareness Month Interactive version of the complete poster is available at: http://mam2000.mathforum.com/ FOCUS March 2000 FOCUS is published by the Mathematical Association of America in January. February. March. April. May/June. August/September. FOCUS October. November. and December. a Editor: Fernando Gouvea. Colby College; March 2000 [email protected] Managing Editor: Carol Baxter. MAA Volume 20. Number 3 [email protected] Senior Writer: Harry Waldman. MAA In This Issue [email protected] Please address advertising inquiries to: 3 "Math Spans All Dimensions" During April Math Awareness Carol Baxter. MAA; [email protected] Month President: Thomas Banchoff. Brown University 3 Felix Browder Named Recipient of National Medal of Science First Vice-President: Barbara Osofsky. By Don Albers Second Vice-President: Frank Morgan. Secretary: Martha Siegel. Treasurer: Gerald 4 Updating the NCTM Standards J. Porter By Kenneth A. Ross Executive Director: Tina Straley 5 A Different Pencil Associate Executive Director and Direc Moving Our Focus from Teachers to Students tor of Publications and Electronic Services: Donald J. Albers By Ed Dubinsky FOCUS Editorial Board: Gerald 6 Mathematics Across the Curriculum at Dartmouth Alexanderson; Donna Beers; J. Kevin By Dorothy I. Wallace Colligan; Ed Dubinsky; Bill Hawkins; Dan Kalman; Maeve McCarthy; Peter Renz; Annie 7 ARUME is the First SIGMAA Selden; Jon Scott; Ravi Vakil. Letters to the editor should be addressed to 8 Read This! Fernando Gouvea. Colby College. Dept. of Mathematics. Waterville. ME 04901. 8 Raoul Bott and Jean-Pierre Serre Share the Wolf Prize Subscription and membership questions 10 Call For Papers should be directed to the MAA Customer Thirteenth Annual MAA Undergraduate Student Paper Sessions Service Center. -
Algebra + Homotopy = Operad
Symplectic, Poisson and Noncommutative Geometry MSRI Publications Volume 62, 2014 Algebra + homotopy = operad BRUNO VALLETTE “If I could only understand the beautiful consequences following from the concise proposition d 2 0.” —Henri Cartan D This survey provides an elementary introduction to operads and to their ap- plications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher homotopies. We try to show how universal this theory is by giving many applications in algebra, geometry, topology, and mathematical physics. (This text is accessible to any student knowing what tensor products, chain complexes, and categories are.) Introduction 229 1. When algebra meets homotopy 230 2. Operads 239 3. Operadic syzygies 253 4. Homotopy transfer theorem 272 Conclusion 283 Acknowledgements 284 References 284 Introduction Galois explained to us that operations acting on the solutions of algebraic equa- tions are mathematical objects as well. The notion of an operad was created in order to have a well defined mathematical object which encodes “operations”. Its name is a portemanteau word, coming from the contraction of the words “operations” and “monad”, because an operad can be defined as a monad encoding operations. The introduction of this notion was prompted in the 60’s, by the necessity of working with higher operations made up of higher homotopies appearing in algebraic topology. Algebra is the study of algebraic structures with respect to isomorphisms. Given two isomorphic vector spaces and one algebra structure on one of them, 229 230 BRUNO VALLETTE one can always define, by means of transfer, an algebra structure on the other space such that these two algebra structures become isomorphic. -
Finite Group Actions on Kervaire Manifolds 3
FINITE GROUP ACTIONS ON KERVAIRE MANIFOLDS DIARMUID CROWLEY AND IAN HAMBLETON M4k+2 Abstract. Let K be the Kervaire manifold: a closed, piecewise linear (PL) mani- fold with Kervaire invariant 1 and the same homology as the product S2k+1 × S2k+1 of M4k+2 spheres. We show that a finite group of odd order acts freely on K if and only if 2k+1 2k+1 it acts freely on S × S . If MK is smoothable, then each smooth structure on M j M4k+2 K admits a free smooth involution. If k 6= 2 − 1, then K does not admit any M30 M62 free TOP involutions. Free “exotic” (PL) involutions are constructed on K , K , and M126 M30 Z Z K . Each smooth structure on K admits a free /2 × /2 action. 1. Introduction One of the main themes in geometric topology is the study of smooth manifolds and their piece-wise linear (PL) triangulations. Shortly after Milnor’s discovery [54] of exotic smooth 7-spheres, Kervaire [39] constructed the first example (in dimension 10) of a PL- manifold with no differentiable structure, and a new exotic smooth 9-sphere Σ9. The construction of Kervaire’s 10-dimensional manifold was generalized to all dimen- sions of the form m ≡ 2 (mod 4), via “plumbing” (see [36, §8]). Let P 4k+2 denote the smooth, parallelizable manifold of dimension 4k+2, k ≥ 0, constructed by plumbing two copies of the the unit tangent disc bundle of S2k+1. The boundary Σ4k+1 = ∂P 4k+2 is a smooth homotopy sphere, now usually called the Kervaire sphere. -
Bibliography
Bibliography [AK98] V. I. Arnold and B. A. Khesin, Topological methods in hydrodynamics, Springer- Verlag, New York, 1998. [AL65] Holt Ashley and Marten Landahl, Aerodynamics of wings and bodies, Addison- Wesley, Reading, MA, 1965, Section 2-7. [Alt55] M. Altman, A generalization of Newton's method, Bulletin de l'academie Polonaise des sciences III (1955), no. 4, 189{193, Cl.III. [Arm83] M.A. Armstrong, Basic topology, Springer-Verlag, New York, 1983. [Bat10] H. Bateman, The transformation of the electrodynamical equations, Proc. Lond. Math. Soc., II, vol. 8, 1910, pp. 223{264. [BB69] N. Balabanian and T.A. Bickart, Electrical network theory, John Wiley, New York, 1969. [BLG70] N. N. Balasubramanian, J. W. Lynn, and D. P. Sen Gupta, Differential forms on electromagnetic networks, Butterworths, London, 1970. [Bos81] A. Bossavit, On the numerical analysis of eddy-current problems, Computer Methods in Applied Mechanics and Engineering 27 (1981), 303{318. [Bos82] A. Bossavit, On finite elements for the electricity equation, The Mathematics of Fi- nite Elements and Applications IV (MAFELAP 81) (J.R. Whiteman, ed.), Academic Press, 1982, pp. 85{91. [Bos98] , Computational electromagnetism: Variational formulations, complementar- ity, edge elements, Academic Press, San Diego, 1998. [Bra66] F.H. Branin, The algebraic-topological basis for network analogies and the vector cal- culus, Proc. Symp. Generalised Networks, Microwave Research, Institute Symposium Series, vol. 16, Polytechnic Institute of Brooklyn, April 1966, pp. 453{491. [Bra77] , The network concept as a unifying principle in engineering and physical sciences, Problem Analysis in Science and Engineering (K. Husseyin F.H. Branin Jr., ed.), Academic Press, New York, 1977. -
Mapping Surgery to Analysis III: Exact Sequences
K-Theory (2004) 33:325–346 © Springer 2005 DOI 10.1007/s10977-005-1554-7 Mapping Surgery to Analysis III: Exact Sequences NIGEL HIGSON and JOHN ROE Department of Mathematics, Penn State University, University Park, Pennsylvania 16802. e-mail: [email protected]; [email protected] (Received: February 2004) Abstract. Using the constructions of the preceding two papers, we construct a natural transformation (after inverting 2) from the Browder–Novikov–Sullivan–Wall surgery exact sequence of a compact manifold to a certain exact sequence of C∗-algebra K-theory groups. Mathematics Subject Classifications (1991): 19J25, 19K99. Key words: C∗-algebras, L-theory, Poincare´ duality, signature operator. This is the final paper in a series of three whose objective is to construct a natural transformation from the surgery exact sequence of Browder, Novikov, Sullivan and Wall [17,21] to a long exact sequence of K-theory groups associated to a certain C∗-algebra extension; we finally achieve this objective in Theorem 5.4. In the first paper [5], we have shown how to associate a homotopy invariant C∗-algebraic signature to suitable chain complexes of Hilbert modules satisfying Poincare´ duality. In the second paper, we have shown that such Hilbert–Poincare´ complexes arise natu- rally from geometric examples of manifolds and Poincare´ complexes. The C∗-algebras that are involved in these calculations are analytic reflections of the equivariant and/or controlled structure of the underlying topology. In paper II [6] we have also clarified the relationship between the analytic signature, defined by the procedure of paper I for suitable Poincare´ com- plexes, and the analytic index of the signature operator, defined only for manifolds. -
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. -
Raoul Bott (1923–2005)
Remembering Raoul Bott (1923–2005) Loring W. Tu, Coordinating Editor With contributions from Rodolfo Gurdian, Stephen Smale, David Mumford, Arthur Jaffe, Shing-Tung Yau, and Loring W. Tu Raoul Bott passed away The contributions are listed in the order in on December 20, 2005. which the contributors first met Raoul Bott. As Over a five-decade career the coordinating editor, I have added a short he made many profound introductory paragraph (in italics) to the beginning and fundamental contri- of each contribution. —Loring Tu butions to geometry and topology. This is the sec- Rodolfo Gurdian ond part of a two-part article in the Notices to Rodolfo Gurdian was one of Raoul Bott’s room- commemorate his life and mates when they were undergraduates at McGill. work. The first part was The imaginary chicken-stealing incident in this arti- an authorized biography, cle is a reference to a real chicken leg incident they “The life and works of experienced together at Mont Tremblant, recounted Raoul Bott” [4], which in [4]. he read and approved What follows is an account of some of the mischief a few years before his that Raoul Bott and I carried out during our days Photo by Bachrach. at McGill. Figure 1. Raoul Bott in 2002. death. Since then there have been at least three I met Raoul in 1941, when we were in our volumes containing remembrances of Raoul Bott first year at McGill University. Both of us lived in by his erstwhile collaborators, colleagues, students, Douglas Hall, a student dormitory of the university, and friends [1], [2], [7]. -
17 Oct 2019 Sir Michael Atiyah, a Knight Mathematician
Sir Michael Atiyah, a Knight Mathematician A tribute to Michael Atiyah, an inspiration and a friend∗ Alain Connes and Joseph Kouneiher Sir Michael Atiyah was considered one of the world’s foremost mathematicians. He is best known for his work in algebraic topology and the codevelopment of a branch of mathematics called topological K-theory together with the Atiyah-Singer index theorem for which he received Fields Medal (1966). He also received the Abel Prize (2004) along with Isadore M. Singer for their discovery and proof of the index the- orem, bringing together topology, geometry and analysis, and for their outstanding role in building new bridges between mathematics and theoretical physics. Indeed, his work has helped theoretical physicists to advance their understanding of quantum field theory and general relativity. Michael’s approach to mathematics was based primarily on the idea of finding new horizons and opening up new perspectives. Even if the idea was not validated by the mathematical criterion of proof at the beginning, “the idea would become rigorous in due course, as happened in the past when Riemann used analytic continuation to justify Euler’s brilliant theorems.” For him an idea was justified by the new links between different problems which it illuminated. Our experience with him is that, in the manner of an explorer, he adapted to the landscape he encountered on the way until he conceived a global vision of the setting of the problem. Atiyah describes here 1 his way of doing mathematics2 : arXiv:1910.07851v1 [math.HO] 17 Oct 2019 Some people may sit back and say, I want to solve this problem and they sit down and say, “How do I solve this problem?” I don’t. -
Council Congratulates Exxon Education Foundation
from.qxp 4/27/98 3:17 PM Page 1315 From the AMS ics. The Exxon Education Foundation funds programs in mathematics education, elementary and secondary school improvement, undergraduate general education, and un- dergraduate developmental education. —Timothy Goggins, AMS Development Officer AMS Task Force Receives Two Grants The AMS recently received two new grants in support of its Task Force on Excellence in Mathematical Scholarship. The Task Force is carrying out a program of focus groups, site visits, and information gathering aimed at developing (left to right) Edward Ahnert, president of the Exxon ways for mathematical sciences departments in doctoral Education Foundation, AMS President Cathleen institutions to work more effectively. With an initial grant Morawetz, and Robert Witte, senior program officer for of $50,000 from the Exxon Education Foundation, the Task Exxon. Force began its work by organizing a number of focus groups. The AMS has now received a second grant of Council Congratulates Exxon $50,000 from the Exxon Education Foundation, as well as a grant of $165,000 from the National Science Foundation. Education Foundation For further information about the work of the Task Force, see “Building Excellence in Doctoral Mathematics De- At the Summer Mathfest in Burlington in August, the AMS partments”, Notices, November/December 1995, pages Council passed a resolution congratulating the Exxon Ed- 1170–1171. ucation Foundation on its fortieth anniversary. AMS Pres- ident Cathleen Morawetz presented the resolution during —Timothy Goggins, AMS Development Officer the awards banquet to Edward Ahnert, president of the Exxon Education Foundation, and to Robert Witte, senior program officer with Exxon. -
Surgery on Compact Manifolds, Second Edition, 1999 68 David A
http://dx.doi.org/10.1090/surv/069 Selected Titles in This Series 69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, Second Edition, 1999 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second Edition, 1999 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P.