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Arxiv:1006.1489V2 [Math.GT] 8 Aug 2010 Ril.Ias Rfie Rmraigtesre Rils[14 Articles Survey the Reading from Profited Also I Article
Pure and Applied Mathematics Quarterly Volume 8, Number 1 (Special Issue: In honor of F. Thomas Farrell and Lowell E. Jones, Part 1 of 2 ) 1—14, 2012 The Work of Tom Farrell and Lowell Jones in Topology and Geometry James F. Davis∗ Tom Farrell and Lowell Jones caused a paradigm shift in high-dimensional topology, away from the view that high-dimensional topology was, at its core, an algebraic subject, to the current view that geometry, dynamics, and analysis, as well as algebra, are key for classifying manifolds whose fundamental group is infinite. Their collaboration produced about fifty papers over a twenty-five year period. In this tribute for the special issue of Pure and Applied Mathematics Quarterly in their honor, I will survey some of the impact of their joint work and mention briefly their individual contributions – they have written about one hundred non-joint papers. 1 Setting the stage arXiv:1006.1489v2 [math.GT] 8 Aug 2010 In order to indicate the Farrell–Jones shift, it is necessary to describe the situation before the onset of their collaboration. This is intimidating – during the period of twenty-five years starting in the early fifties, manifold theory was perhaps the most active and dynamic area of mathematics. Any narrative will have omissions and be non-linear. Manifold theory deals with the classification of ∗I thank Shmuel Weinberger and Tom Farrell for their helpful comments on a draft of this article. I also profited from reading the survey articles [14] and [4]. 2 James F. Davis manifolds. There is an existence question – when is there a closed manifold within a particular homotopy type, and a uniqueness question, what is the classification of manifolds within a homotopy type? The fifties were the foundational decade of manifold theory. -
Stable Real Cohomology of Arithmetic Groups
ANNALES SCIENTIFIQUES DE L’É.N.S. ARMAND BOREL Stable real cohomology of arithmetic groups Annales scientifiques de l’É.N.S. 4e série, tome 7, no 2 (1974), p. 235-272 <http://www.numdam.org/item?id=ASENS_1974_4_7_2_235_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1974, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. sclent. EC. Norm. Sup., 46 serie, t. 7, 1974, p. 235 a 272. STABLE REAL COHOMOLOGY OF ARITHMETIC GROUPS BY ARMAND BOREL A Henri Cartan, a Poccasion de son 70® anniversaire Let r be an arithmetic subgroup of a semi-simple group G defined over the field of rational numbers Q. The real cohomology H* (T) of F may be identified with the coho- mology of the complex Q^ of r-invariant smooth differential forms on the symmetric space X of maximal compact subgroups of the group G (R) of real points of G. Let 1^ be the space of differential forms on X which are invariant under the identity component G (R)° of G (R). It is well-known to consist of closed (in fact harmonic) forms, whence a natural homomorphism^* : 1^ —> H* (T). -
Algebraic D-Modules and Representation Theory Of
Contemporary Mathematics Volume 154, 1993 Algebraic -modules and Representation TheoryDof Semisimple Lie Groups Dragan Miliˇci´c Abstract. This expository paper represents an introduction to some aspects of the current research in representation theory of semisimple Lie groups. In particular, we discuss the theory of “localization” of modules over the envelop- ing algebra of a semisimple Lie algebra due to Alexander Beilinson and Joseph Bernstein [1], [2], and the work of Henryk Hecht, Wilfried Schmid, Joseph A. Wolf and the author on the localization of Harish-Chandra modules [7], [8], [13], [17], [18]. These results can be viewed as a vast generalization of the classical theorem of Armand Borel and Andr´e Weil on geometric realiza- tion of irreducible finite-dimensional representations of compact semisimple Lie groups [3]. 1. Introduction Let G0 be a connected semisimple Lie group with finite center. Fix a maximal compact subgroup K0 of G0. Let g be the complexified Lie algebra of G0 and k its subalgebra which is the complexified Lie algebra of K0. Denote by σ the corresponding Cartan involution, i.e., σ is the involution of g such that k is the set of its fixed points. Let K be the complexification of K0. The group K has a natural structure of a complex reductive algebraic group. Let π be an admissible representation of G0 of finite length. Then, the submod- ule V of all K0-finite vectors in this representation is a finitely generated module over the enveloping algebra (g) of g, and also a direct sum of finite-dimensional U irreducible representations of K0. -
Mathematics People
Mathematics People Hong Kong University of Science and Technology, and Awards Presented at 2007 Mu-Tao Wang, Columbia University. ICCM The ICCM International Cooperation Award is presented to an individual who has promoted the development of At each International Congress of Chinese Mathemati- mathematics in China, Hong Kong, and Taiwan through cians (ICCM), the winners of several prestigious awards collaboration, teaching, and support of Chinese mathema- are announced during the opening ceremony. These ticians. The inaugural award was presented in 2004. The awards include the Morningside Medal of Mathematics, the 2007 ICCM International Cooperation Award is awarded to Chern Prize in Mathematics, and the ICCM International Stanley Osher, University of California at Los Angeles. Cooperation Award. The Fourth ICCM was held in Hang- Supported by the New World Development Company zhou, China, December 17–22, 2007. At ICCM 2007, two Ltd., the New World Mathematics Awards recognize out- new prizes were introduced: the New World Mathematics standing doctoral, master’s, and undergraduate theses Awards and the S. T. Yau Mathematics Awards. written by mathematicians of Chinese descent who have The Morningside Medal of Mathematics is awarded to graduated from universities and institutes in the past exceptional mathematicians of Chinese descent under three years. The purpose is to provide encouragement the age of forty-five for their seminal achievements in to talented Chinese mathematicians and to promote cre- mathematics and applied mathematics. The winners of ativity and innovation in mathematics. Six Ph.D. Thesis the Morningside Medal of Mathematics are tradition- Awards, five Master Thesis Awards, and ten Bachelor ally announced at the ICCM. -
Fifth International Congress of Chinese Mathematicians Part 1
AMS/IP Studies in Advanced Mathematics S.-T. Yau, Series Editor Fifth International Congress of Chinese Mathematicians Part 1 Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau Editors American Mathematical Society • International Press Fifth International Congress of Chinese Mathematicians https://doi.org/10.1090/amsip/051.1 AMS/IP Studies in Advanced Mathematics Volume 51, Part 1 Fifth International Congress of Chinese Mathematicians Lizhen Ji Yat Sun Poon Lo Yang Shing-Tung Yau Editors American Mathematical Society • International Press Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 05–XX, 08–XX, 11–XX, 14–XX, 22–XX, 35–XX, 37–XX, 53–XX, 58–XX, 62–XX, 65–XX, 20–XX, 30–XX, 80–XX, 83–XX, 90–XX. All photographs courtesy of International Press. Library of Congress Cataloging-in-Publication Data International Congress of Chinese Mathematicians (5th : 2010 : Beijing, China) p. cm. (AMS/IP studies in advanced mathematics ; v. 51) Includes bibliographical references. ISBN 978-0-8218-7555-1 (set : alk. paper)—ISBN 978-0-8218-7586-5 (pt. 1 : alk. paper)— ISBN 978-0-8218-7587-2 (pt. 2 : alk. paper) 1. Mathematics—Congresses. I. Ji, Lizhen, 1964– II. Title. III. Title: 5th International Congress of Chinese Mathematicians. QA1.I746 2010 510—dc23 2011048032 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. -
Issue 118 ISSN 1027-488X
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY S E European M M Mathematical E S Society December 2020 Issue 118 ISSN 1027-488X Obituary Sir Vaughan Jones Interviews Hillel Furstenberg Gregory Margulis Discussion Women in Editorial Boards Books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Recent books in the EMS Monographs in Mathematics series Massimiliano Berti (SISSA, Trieste, Italy) and Philippe Bolle (Avignon Université, France) Quasi-Periodic Solutions of Nonlinear Wave Equations on the d-Dimensional Torus 978-3-03719-211-5. 2020. 374 pages. Hardcover. 16.5 x 23.5 cm. 69.00 Euro Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrödinger equation, can be viewed as infinite-dimensional Hamiltonian systems. In the last thirty years, several existence results of time quasi-periodic solutions have been proved adopting a “dynamical systems” point of view. Most of them deal with equations in one space dimension, whereas for multidimensional PDEs a satisfactory picture is still under construction. An updated introduction to the now rich subject of KAM theory for PDEs is provided in the first part of this research monograph. We then focus on the nonlinear wave equation, endowed with periodic boundary conditions. The main result of the monograph proves the bifurcation of small amplitude finite-dimensional invariant tori for this equation, in any space dimension. -
Armand Borel (1923–2003), Volume 51, Number 5
Armand Borel (1923–2003) James Arthur, Enrico Bombieri, Komaravolu Chandrasekharan, Friedrich Hirzebruch, Gopal Prasad, Jean-Pierre Serre, Tonny A. Springer, and Jacques Tits Citations within this article. Borel’s Collected Papers are [Œ], and his 17 books are referred to as [1] through [17]. These are all listed in a sidebar on page 501. An item like [Œ 23] is Borel paper number 23 in [Œ]. Ci- tations of work by people other than Borel are by letter combinations such as [Che], and the de- tails appear at the end of the article. their root systems. He spent the year 1949–50 in 1 Institute for Advanced Study, 1999. Paris, with a grant from the CNRS . A good choice Armand Borel (for us, as well as for him), Paris being the very spot where what Americans have called “French Topol- Jean-Pierre Serre ogy” was being created, with the courses from The Swiss mathematician Armand Borel died Leray at the Collège de France and the Cartan sem- August 11, 2003, in Princeton from a rapidly evolv- inar at the École Normale Supérieure. Borel was an ing cancer. Few foreign mathematicians had as active participant in the Cartan seminar while many connections with France. He was a student closely following Leray’s courses. He managed to of Leray, he took part in the Cartan seminar, and understand the famous “spectral sequence”, not an he published more than twenty papers in collabo- easy task, and he explained it to me so well that I ration with our colleagues Lichnerowicz and Tits, have not stopped using it since. -
Lie Groups and Linear Algebraic Groups I. Complex and Real Groups Armand Borel
Lie Groups and Linear Algebraic Groups I. Complex and Real Groups Armand Borel 1. Root systems x 1.1. Let V be a finite dimensional vector space over Q. A finite subset of V is a root system if it satisfies: RS 1. Φ is finite, consists of non-zero elements and spans V . RS 2. Given a Φ, there exists an automorphism ra of V preserving Φ 2 ra such that ra(a) = a and its fixed point set V has codimension 1. [Such a − transformation is unique, of order 2.] The Weyl group W (Φ) or W of Φ is the subgroup of GL(V ) generated by the ra (a Φ). It is finite. Fix a positive definite scalar product ( , ) on V invariant 2 under W . Then ra is the reflection to the hyperplane a. ? 1 RS 3. Given u; v V , let nu;v = 2(u; v) (v; v)− . We have na;b Z for all 2 · 2 a; b Φ. 2 1.2. Some properties. (a) If a and c a (c > 0) belong to Φ, then c = 1; 2. · The system Φ is reduced if only c = 1 occurs. (b) The reflection to the hyperplane a = 0 (for any a = 0) is given by 6 (1) ra(v) = v nv;aa − therefore if a; b Φ are linearly independent, and (a; b) > 0 (resp. (a; b) < 0), 2 then a b (resp. a + b) is a root. On the other hand, if (a; b) = 0, then either − a + b and a b are roots, or none of them is (in which case a and b are said to be − strongly orthogonal). -
Lie Groups and Automorphic Forms This Page Intentionally Left Blank AMS/IP Studies in Advanced Mathematics
Lie Groups and Automorphic Forms This page intentionally left blank https://doi.org/10.1090/amsip/037 AMS/IP Studies in Advanced Mathematics Volume 37 Lie Groups and Automorphic Forms Proceedings of the 2003 Summer Program Zhejiang University Center of Mathematical Sciences Hangzhou, China Lizhen Ji, Managing Editor Jian-Shu Li, H. W. Xu, and Shing-Tung Yau, Editors American Mathematical Society • International Press Shing-Tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 20Gxx, 22Exx, 11-XX, 55Nxx. The paper, "On the cohomology of locally symmetric spaces and of their compactifi- cations," by Leslie Saper, starting on page 169, is used by permission. © 2002, Current Developments in Mathematics, International Press, Somerville, MA. The photograph of Armand Borel on page v is courtesy of Armand Borel's daughter, Dominique Borel. Library of Congress Cataloging-in-Publication Data Lie groups and automorphic forms / [Lizhen Ji, editor]. p. cm. — (Studies in advanced mathematics, ISSN 1089-3288 ; v. 37) Includes bibliographical references. ISBN-13: 978-0-8218-4198-3 (alk. paper) ISBN-10: 0-8218-4198-X (alk. paper) 1. Lie groups. 2. Automorphic forms. I. Ji, Lizhen, 1964- QA387.L52 2006 512'.482—dc22 2006048421 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. -
Armand Borel 1923–2003
Armand Borel 1923–2003 A Biographical Memoir by Mark Goresky ©2019 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. ARMAND BOREL May 21, 1923–August 11, 2003 Elected to the NAS, 1987 Armand Borel was a leading mathematician of the twen- tieth century. A native of Switzerland, he spent most of his professional life in Princeton, New Jersey, where he passed away after a short illness in 2003. Although he is primarily known as one of the chief archi- tects of the modern theory of linear algebraic groups and of arithmetic groups, Borel had an extraordinarily wide range of mathematical interests and influence. Perhaps more than any other mathematician in modern times, Borel elucidated and disseminated the work of others. family the Borel of Photo courtesy His books, conference proceedings, and journal publica- tions provide the document of record for many important mathematical developments during his lifetime. By Mark Goresky Mathematical objects and results bearing Borel’s name include Borel subgroups, Borel regulator, Borel construction, Borel equivariant cohomology, Borel-Serre compactification, Bailey- Borel compactification, Borel fixed point theorem, Borel-Moore homology, Borel-Weil theorem, Borel-de Siebenthal theorem, and Borel conjecture.1 Borel was awarded the Brouwer Medal (Dutch Mathematical Society), the Steele Prize (American Mathematical Society) and the Balzan Prize (Italian-Swiss International Balzan Foundation). He was a member of the National Academy of Sciences (USA), the American Academy of Arts and Sciences, the American Philosophical Society, the Finnish Academy of Sciences and Letters, the Academia Europa, and the French Academy of Sciences.1 1 Borel enjoyed pointing out, with some amusement, that he was not related to the famous Borel of “Borel sets." 2 ARMAND BOREL Switzerland and France Armand Borel was born in 1923 in the French-speaking city of La Chaux-de-Fonds in Switzerland. -
Variations on a Theme of Borel
VARIATIONS ON A THEME OF BOREL: AN ESSAY CONCERNING THE ROLE OF THE FUNDAMENTAL GROUP IN RIGIDITY Revision Summer 2019 Shmuel Weinberger University of Chicago Unintentionally Blank 2 Armand Borel William Thurston 3 4 Preface. This essay is a work of historical fiction – the “What if Eleanor Roosevelt could fly?” kind1. The Borel conjecture is a central problem in topology: it asserts the topological rigidity of aspherical manifolds (definitions below!). Borel made his conjecture in a letter to Serre some 65 years ago2, after learning of some work of Mostow on the rigidity of solvmanifolds. We shall re-imagine Borel’s conjecture as being made after Mostow had proved the more famous rigidity theorem that bears his name – the rigidity of hyperbolic manifolds of dimension at least three – as the geometric rigidity of hyperbolic manifolds is stronger than what is true of solvmanifolds, and the geometric picture is clearer. I will consider various related problems in a completely ahistorical order. My motive in all this is to highlight and explain various ideas, especially recurring ideas, that illuminate our (or at least my own) current understanding of this area. Based on the analogy between geometry and topology imagined by Borel, one can make many other conjectures: variations on Borel’s theme. Many, but perhaps not all, of these variants are false and one cannot blame them on Borel. (On several occasions he described feeling lucky that he ducked the bullet and had not conjectured smooth rigidity – a phenomenon indistinguishable to the mathematics of the time from the statement that he did conjecture.) However, even the false variants are false for good reasons and studying these can quite fun (and edifying); all of the problems we consider enrich our understanding of the geometric and analytic properties of manifolds. -
Furstenberg and Margulis Awarded 2020 Abel Prize
COMMUNICATION Furstenberg and Margulis Awarded 2020 Abel Prize The Norwegian Academy of Science and Letters has awarded the Abel Prize for 2020 to Hillel Furstenberg of the Hebrew University of Jerusalem and Gregory Margulis of Yale University “for pioneering the use of methods from probability and dynamics in group theory, number theory, and combinatorics.” Starting from the study of random products of matrices, in 1963, Hillel Furstenberg introduced and classified a no- tion of fundamental importance, now called “Furstenberg boundary.” Using this, he gave a Poisson-type formula expressing harmonic functions on a general group in terms of their boundary values. In his works on random walks at the beginning of the 1960s, some in collaboration with Harry Kesten, he also obtained an important criterion for the positivity of the largest Lyapunov exponent. Motivated by Diophantine approximation, in 1967, Furstenberg introduced the notion of disjointness of er- godic systems, a notion akin to that of being coprime for Hillel Furstenberg Gregory Margulis integers. This natural notion turned out to be extremely deep and to have applications to a wide range of areas, Citation including signal processing and filtering questions in A central branch of probability theory is the study of ran- electrical engineering, the geometry of fractal sets, homoge- dom walks, such as the route taken by a tourist exploring neous flows, and number theory. His “×2 ×3 conjecture” is an unknown city by flipping a coin to decide between a beautifully simple example which has led to many further turning left or right at every cross. Hillel Furstenberg and developments.