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THE RIEMANN HYPOTHESIS BERNARD RUSSO University Of
THE RIEMANN HYPOTHESIS BERNARD RUSSO University of California, Irvine PART I THE PRIME NUMBER THEOREM JULY 29, 2010 PACIFIC SUMMER UNSOLVED PROBLEMS SEMINAR PART II THE RIEMANN HYPOTHESIS SEPTEMBER 14, 2010 MATH COLLOQUIUM FULLERTON COLLEGE DEPARTMENT OF MATHEMATICS BEGINNING OF PART I “book report” “PRIME OBSESSION” JOHN DERBYSHIRE 2003 PRESENTATION OF CREDENTIALS Bernard Russo 1965 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 H. A. Schwarz 1864 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 H. A. Schwarz 1864 Karl Weierstrass 1854—E. E. Kummer 1831 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 H. A. Schwarz 1864 Karl Weierstrass 1854—E. E. Kummer 1831 C. Gudermann 1841—H. Scherk 1823 PRESENTATION OF CREDENTIALS Bernard Russo 1965 Henry Dye 1950 Irving Segal 1940 Einar Hille 1918 M. Riesz 1912 L. Fejer 1902 H. A. Schwarz 1864 Karl Weierstrass 1854—E. E. Kummer 1831 C. Gudermann 1841—H. -
A Short Survey of Cyclic Cohomology
Clay Mathematics Proceedings Volume 10, 2008 A Short Survey of Cyclic Cohomology Masoud Khalkhali Dedicated with admiration and affection to Alain Connes Abstract. This is a short survey of some aspects of Alain Connes’ contribu- tions to cyclic cohomology theory in the course of his work on noncommutative geometry over the past 30 years. Contents 1. Introduction 1 2. Cyclic cohomology 3 3. From K-homology to cyclic cohomology 10 4. Cyclic modules 13 5. Thelocalindexformulaandbeyond 16 6. Hopf cyclic cohomology 21 References 28 1. Introduction Cyclic cohomology was discovered by Alain Connes no later than 1981 and in fact it was announced in that year in a conference in Oberwolfach [5]. I have reproduced the text of his abstract below. As it appears in his report, one of arXiv:1008.1212v1 [math.OA] 6 Aug 2010 Connes’ main motivations to introduce cyclic cohomology theory came from index theory on foliated spaces. Let (V, ) be a compact foliated manifold and let V/ denote the space of leaves of (V, ).F This space, with its natural quotient topology,F is, in general, a highly singular spaceF and in noncommutative geometry one usually replaces the quotient space V/ with a noncommutative algebra A = C∗(V, ) called the foliation algebra of (V,F ). It is the convolution algebra of the holonomyF groupoid of the foliation and isF a C∗-algebra. It has a dense subalgebra = C∞(V, ) which plays the role of the algebra of smooth functions on V/ . LetA D be a transversallyF elliptic operator on (V, ). The analytic index of D, index(F D) F ∈ K0(A), is an element of the K-theory of A. -
Noncommutative Geometry and the Spectral Model of Space-Time
S´eminaire Poincar´eX (2007) 179 – 202 S´eminaire Poincar´e Noncommutative geometry and the spectral model of space-time Alain Connes IHES´ 35, route de Chartres 91440 Bures-sur-Yvette - France Abstract. This is a report on our joint work with A. Chamseddine and M. Marcolli. This essay gives a short introduction to a potential application in physics of a new type of geometry based on spectral considerations which is convenient when dealing with noncommutative spaces i.e. spaces in which the simplifying rule of commutativity is no longer applied to the coordinates. Starting from the phenomenological Lagrangian of gravity coupled with matter one infers, using the spectral action principle, that space-time admits a fine structure which is a subtle mixture of the usual 4-dimensional continuum with a finite discrete structure F . Under the (unrealistic) hypothesis that this structure remains valid (i.e. one does not have any “hyperfine” modification) until the unification scale, one obtains a number of predictions whose approximate validity is a basic test of the approach. 1 Background Our knowledge of space-time can be summarized by the transition from the flat Minkowski metric ds2 = − dt2 + dx2 + dy2 + dz2 (1) to the Lorentzian metric 2 µ ν ds = gµν dx dx (2) of curved space-time with gravitational potential gµν . The basic principle is the Einstein-Hilbert action principle Z 1 √ 4 SE[ gµν ] = r g d x (3) G M where r is the scalar curvature of the space-time manifold M. This action principle only accounts for the gravitational forces and a full account of the forces observed so far requires the addition of new fields, and of corresponding new terms SSM in the action, which constitute the Standard Model so that the total action is of the form, S = SE + SSM . -
Space and Time Dimensions of Algebras With
Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and quantum electrodynamics Nadir Bizi, Christian Brouder, Fabien Besnard To cite this version: Nadir Bizi, Christian Brouder, Fabien Besnard. Space and time dimensions of algebras with appli- cations to Lorentzian noncommutative geometry and quantum electrodynamics. Journal of Mathe- matical Physics, American Institute of Physics (AIP), 2018, 59 (6), pp.062303. 10.1063/1.4986228. hal-01398231v2 HAL Id: hal-01398231 https://hal.archives-ouvertes.fr/hal-01398231v2 Submitted on 16 May 2019 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. Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics Nadir Bizi, Christian Brouder, and Fabien Besnard Citation: Journal of Mathematical Physics 59, 062303 (2018); doi: 10.1063/1.5010424 View online: https://doi.org/10.1063/1.5010424 View Table of Contents: http://aip.scitation.org/toc/jmp/59/6 Published by the American Institute of Physics Articles you may -
Noncommutative Geometry and Flavour Mixing
Noncommutative geometry and flavour mixing prepared by Jose´ M. Gracia-Bond´ıa Department of Theoretical Physics Universidad de Zaragoza, 50009 Zaragoza, Spain October 22, 2013 1 Introduction: the universality problem “The origin of the quark and lepton masses is shrouded in mystery” [1]. Some thirty years ago, attempts to solve the enigma based on textures of the quark mass matrices, purposedly reflecting mass hierarchies and “nearest-neighbour” interactions, were very popular. Now, in the late eighties, Branco, Lavoura and Mota [2] showed that, within the SM, the zero pattern 0a b 01 @c 0 dA; (1) 0 e 0 a central ingredient of Fritzsch’s well-known Ansatz for the mass matrices, is devoid of any particular physical meaning. (The top quark is above on top.) Although perhaps this was not immediately clear at the time, paper [2] marked a water- shed in the theory of flavour mixing. In algebraic terms, it establishes that the linear subspace of matrices of the form (1) is universal for the group action of unitaries effecting chiral basis transformations, that respect the charged-current term of the Lagrangian. That is, any mass matrix can be transformed to that form without modifying the corresponding CKM matrix. To put matters in perspective, consider the unitary group acting by similarity on three-by- three matrices. The classical triangularization theorem by Schur ensures that the zero patterns 0a 0 01 0a b c1 @b c 0A; @0 d eA (2) d e f 0 0 f are universal in this sense. However, proof that the zero pattern 0a b 01 @0 c dA e 0 f 1 is universal was published [3] just three years ago! (Any off-diagonal n(n−1)=2 zero pattern with zeroes at some (i j) and no zeroes at the matching ( ji), is universal in this sense, for complex n × n matrices.) Fast-forwarding to the present time, notwithstanding steady experimental progress [4] and a huge amount of theoretical work by many authors, we cannot be sure of being any closer to solving the “Meroitic” problem [5] of divining the spectrum behind the known data. -
Irving Kaplansky
Portraying and remembering Irving Kaplansky Hyman Bass University of Michigan Mathematical Sciences Research Institute • February 23, 2007 1 Irving (“Kap”) Kaplansky “infinitely algebraic” “I liked the algebraic way of looking at things. I’m additionally fascinated when the algebraic method is applied to infinite objects.” 1917 - 2006 A Gallery of Portraits 2 Family portrait: Kap as son • Born 22 March, 1917 in Toronto, (youngest of 4 children) shortly after his parents emigrated to Canada from Poland. • Father Samuel: Studied to be a rabbi in Poland; worked as a tailor in Toronto. • Mother Anna: Little schooling, but enterprising: “Health Bread Bakeries” supported (& employed) the whole family 3 Kap’s father’s grandfather Kap’s father’s parents Kap (age 4) with family 4 Family Portrait: Kap as father • 1951: Married Chellie Brenner, a grad student at Harvard Warm hearted, ebullient, outwardly emotional (unlike Kap) • Three children: Steven, Alex, Lucy "He taught me and my brothers a lot, (including) what is really the most important lesson: to do the thing you love and not worry about making money." • Died 25 June, 2006, at Steven’s home in Sherman Oaks, CA Eight months before his death he was still doing mathematics. Steven asked, -“What are you working on, Dad?” -“It would take too long to explain.” 5 Kap & Chellie marry 1951 Family portrait, 1972 Alex Steven Lucy Kap Chellie 6 Kap – The perfect accompanist “At age 4, I was taken to a Yiddish musical, Die Goldene Kala. It was a revelation to me that there could be this kind of entertainment with music. -
Noncommutative Stacks
Noncommutative Stacks Introduction One of the purposes of this work is to introduce a noncommutative analogue of Artin’s and Deligne-Mumford algebraic stacks in the most natural and sufficiently general way. We start with quasi-coherent modules on fibered categories, then define stacks and prestacks. We define formally smooth, formally unramified, and formally ´etale cartesian functors. This provides us with enough tools to extend to stacks the glueing formalism we developed in [KR3] for presheaves and sheaves of sets. Quasi-coherent presheaves and sheaves on a fibered category. Quasi-coherent sheaves on geometric (i.e. locally ringed topological) spaces were in- troduced in fifties. The notion of quasi-coherent modules was extended in an obvious way to ringed sites and toposes at the moment the latter appeared (in SGA), but it was not used much in this generality. Recently, the subject was revisited by D. Orlov in his work on quasi-coherent sheaves in commutative an noncommutative geometry [Or] and by G. Laumon an L. Moret-Bailly in their book on algebraic stacks [LM-B]. Slightly generalizing [R4], we associate with any functor F (regarded as a category over a category) the category of ’quasi-coherent presheaves’ on F (otherwise called ’quasi- coherent presheaves of modules’ or simply ’quasi-coherent modules’) and study some basic properties of this correspondence in the case when the functor defines a fibered category. Imitating [Gir], we define the quasi-topology of 1-descent (or simply ’descent’) and the quasi-topology of 2-descent (or ’effective descent’) on the base of a fibered category (i.e. -
Noncommutative Geometry, the Spectral Standpoint Arxiv
Noncommutative Geometry, the spectral standpoint Alain Connes October 24, 2019 In memory of John Roe, and in recognition of his pioneering achievements in coarse geometry and index theory. Abstract We update our Year 2000 account of Noncommutative Geometry in [68]. There, we de- scribed the following basic features of the subject: I The natural “time evolution" that makes noncommutative spaces dynamical from their measure theory. I The new calculus which is based on operators in Hilbert space, the Dixmier trace and the Wodzicki residue. I The spectral geometric paradigm which extends the Riemannian paradigm to the non- commutative world and gives a new tool to understand the forces of nature as pure gravity on a more involved geometric structure mixture of continuum with the discrete. I The key examples such as duals of discrete groups, leaf spaces of foliations and de- formations of ordinary spaces, which showed, very early on, that the relevance of this new paradigm was going far beyond the framework of Riemannian geometry. Here, we shall report1 on the following highlights from among the many discoveries made since 2000: 1. The interplay of the geometry with the modular theory for noncommutative tori. 2. Great advances on the Baum-Connes conjecture, on coarse geometry and on higher in- dex theory. 3. The geometrization of the pseudo-differential calculi using smooth groupoids. 4. The development of Hopf cyclic cohomology. 5. The increasing role of topological cyclic homology in number theory, and of the lambda arXiv:1910.10407v1 [math.QA] 23 Oct 2019 operations in archimedean cohomology. 6. The understanding of the renormalization group as a motivic Galois group. -
Annals of K-Theory Vol. 1 (2016)
ANNALSOF K-THEORY Paul Balmer Spencer Bloch vol. 1 no. 4 2016 Alain Connes Guillermo Cortiñas Eric Friedlander Max Karoubi Gennadi Kasparov Alexander Merkurjev Amnon Neeman Jonathan Rosenberg Marco Schlichting Andrei Suslin Vladimir Voevodsky Charles Weibel Guoliang Yu msp A JOURNAL OF THE K-THEORY FOUNDATION ANNALS OF K-THEORY msp.org/akt EDITORIAL BOARD Paul Balmer University of California, Los Angeles, USA [email protected] Spencer Bloch University of Chicago, USA [email protected] Alain Connes Collège de France; Institut des Hautes Études Scientifiques; Ohio State University [email protected] Guillermo Cortiñas Universidad de Buenos Aires and CONICET, Argentina [email protected] Eric Friedlander University of Southern California, USA [email protected] Max Karoubi Institut de Mathématiques de Jussieu – Paris Rive Gauche, France [email protected] Gennadi Kasparov Vanderbilt University, USA [email protected] Alexander Merkurjev University of California, Los Angeles, USA [email protected] Amnon Neeman amnon.Australian National University [email protected] Jonathan Rosenberg (Managing Editor) University of Maryland, USA [email protected] Marco Schlichting University of Warwick, UK [email protected] Andrei Suslin Northwestern University, USA [email protected] Vladimir Voevodsky Institute for Advanced Studies, USA [email protected] Charles Weibel (Managing Editor) Rutgers University, USA [email protected] Guoliang Yu Texas A&M University, USA [email protected] PRODUCTION Silvio Levy (Scientific Editor) [email protected] Annals of K-Theory is a journal of the K-Theory Foundation(ktheoryfoundation.org). The K-Theory Foundation acknowledges the precious support of Foundation Compositio Mathematica, whose help has been instrumental in the launch of the Annals of K-Theory. -
Irving Ezra Segal (1918–1998)
mem-segal.qxp 5/12/99 12:57 PM Page 659 Irving Ezra Segal (1918–1998) John C. Baez, Edwin F. Beschler, Leonard Gross, Bertram Kostant, Edward Nelson, Michèle Vergne, and Arthur S. Wightman Irving Segal died suddenly on August 30, 1998, After the war while taking an evening walk. He was seventy-nine Segal spent two and was vigorously engaged in research. years at the Insti- Born on September 13, 1918, in the Bronx, he tute for Advanced grew up in Trenton and received his A.B. from Study, where he Princeton in 1937. What must it have been like to held the first of the be a member of the Jewish quota at Princeton in three Guggenheim the 1930s? He told me once that a fellow under- Fellowships that he graduate offered him money to take an exam in his was to win. Other stead and was surprised when Irving turned him honors included down. election to the Na- He received his Ph.D. from Yale in 1940. His the- tional Academy of sis was written under the nominal direction of Sciences in 1973 Einar Hille, who suggested that Segal continue his and the Humboldt Award in 1981. At and Tamarkin’s investigation of the ideal theory the University of of the algebra of Laplace-Stieltjes transforms ab- Chicago from 1948 solutely convergent in a fixed half-plane. But, Segal to 1960, he had fif- wrote, “For conceptual clarification and for other teen doctoral stu- reasons, an investigation of the group algebra of dents, and at MIT, a general [locally compact] abelian group was of where he was pro- Irving Segal interest.” And the thesis was not restricted to fessor from 1960 abelian groups. -
Lie Groupoids, Pseudodifferential Calculus and Index Theory
Lie groupoids, pseudodifferential calculus and index theory by Claire Debord and Georges Skandalis Université Paris Diderot, Sorbonne Paris Cité Sorbonne Universités, UPMC Paris 06, CNRS, IMJ-PRG UFR de Mathématiques, CP 7012 - Bâtiment Sophie Germain 5 rue Thomas Mann, 75205 Paris CEDEX 13, France [email protected] [email protected] Abstract Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pio- neering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C∗-algebras, their pseudodifferential calculus... We review several re- cent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject. Contents 1 Introduction 2 2 Lie groupoids and their operators algebras 3 2.1 Lie groupoids . .3 2.1.1 Generalities . .3 2.1.2 Morita equivalence of Lie groupoids . .7 2.2 C∗-algebra of a Lie groupoid . .7 2.2.1 Convolution ∗-algebra of smooth functions with compact support . .7 2.2.2 Norm and C∗-algebra . .8 2.3 Deformation to the normal cone and blowup groupoids . .9 2.3.1 Deformation to the normal cone groupoid . .9 2.3.2 Blowup groupoid . 10 3 Pseudodifferential calculus on Lie groupoids 11 3.1 Distributions on G conormal to G(0) ........................... 12 3.1.1 Symbols and conormal distributions . 12 3.1.2 Convolution . 15 3.1.3 Pseudodifferential operators of order ≤ 0 ..................... 16 3.1.4 Analytic index . 17 3.2 Classical examples . -
On the Generalized Non-Commutative Sphere and Their K-Theory
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1325-1335 © Research India Publications http://www.ripublication.com On the generalized non-commutative sphere and their K-theory Saleh Omran Taif University, Faculty of Science, Taif , KSA South valley university, Faculty of science, Math. Dep. Qena , Egypt. Abstract In the work we follow the work of J.Cuntz, he associate a universal C* - algebra to every locally finite flag simplicial complex. In this article we define * nc a universal C -algebra S3 , associated to the 3 -dimensional noncommutative sphere. We analyze the topological information of the noncommutative sphere by using it’s skeleton filtration Ik )( . We examine the nc K -theory of the quotient 3 /IS k , and I k for such kk 7, . Keywords: Universal C* -algebra , Noncommutative sphere, K -theory of C* -algebra. 2000 AMS : 19 K 46. 1. Introduction Noncommutative 3 -sphere play an important role in the non commutative geometry which introduced by Allain Connes [4] .In noncommutative geometry, the set of points in the space are replaced by the set of continuous functions on the space. In fact noncommutative geometry change our point of view from the topological space itself to the functions on the space ( The algebra of functions on the space). Indeed, the main idea was noticed first by Gelfand-Niemark [8] they were give an important theorem which state that: any commutative C* -algebra is isomorphic of the continuous functions of a compact space, namely the space of characters of the algebra. More generally any C* -algebra ( commutative or not) is isomorphic to a closed subalgebra of the algebra of bounded operators on a Hilbert space.