K-Theory for Group C -Algebras
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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. -
MA3403 Algebraic Topology Lecturer: Gereon Quick Lecture 21
MA3403 Algebraic Topology Lecturer: Gereon Quick Lecture 21 21. Applications of cup products in cohomology We are going to see some examples where we calculate or apply multiplicative structures on cohomology. But we start with a couple of facts we forgot to mention last time. Relative cup products Let (X;A) be a pair of spaces. The formula which specifies the cup product by its effect on a simplex (' [ )(σ) = '(σj[e0;:::;ep]) (σj[ep;:::;ep+q]) extends to relative cohomology. For, if σ : ∆p+q ! X has image in A, then so does any restriction of σ. Thus, if either ' or vanishes on chains with image in A, then so does ' [ . Hence we get relative cup product maps Hp(X; R) × Hq(X;A; R) ! Hp+q(X;A; R) Hp(X;A; R) × Hq(X; R) ! Hp+q(X;A; R) Hp(X;A; R) × Hq(X;A; R) ! Hp+q(X;A; R): More generally, assume we have two open subsets A and B of X. Then the formula for ' [ on cochains implies that cup product yields a map Sp(X;A; R) × Sq(X;B; R) ! Sp+q(X;A + B; R) where Sn(X;A+B; R) denotes the subgroup of Sn(X; R) of cochains which vanish on sums of chains in A and chains in B. The natural inclusion Sn(X;A [ B; R) ,! Sn(X;A + B; R) induces an isomorphism in cohomology. For we have a map of long exact coho- mology sequences Hn(A [ B) / Hn(X) / Hn(X;A [ B) / Hn+1(A [ B) / Hn+1(X) Hn(A + B) / Hn(X) / Hn(X;A + B) / Hn+1(A + B) / Hn+1(X) 1 2 where we omit the coefficients. -
Homological Mirror Symmetry for the Genus 2 Curve in an Abelian Variety and Its Generalized Strominger-Yau-Zaslow Mirror by Cath
Homological mirror symmetry for the genus 2 curve in an abelian variety and its generalized Strominger-Yau-Zaslow mirror by Catherine Kendall Asaro Cannizzo A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Denis Auroux, Chair Professor David Nadler Professor Marjorie Shapiro Spring 2019 Homological mirror symmetry for the genus 2 curve in an abelian variety and its generalized Strominger-Yau-Zaslow mirror Copyright 2019 by Catherine Kendall Asaro Cannizzo 1 Abstract Homological mirror symmetry for the genus 2 curve in an abelian variety and its generalized Strominger-Yau-Zaslow mirror by Catherine Kendall Asaro Cannizzo Doctor of Philosophy in Mathematics University of California, Berkeley Professor Denis Auroux, Chair Motivated by observations in physics, mirror symmetry is the concept that certain mani- folds come in pairs X and Y such that the complex geometry on X mirrors the symplectic geometry on Y . It allows one to deduce information about Y from known properties of X. Strominger-Yau-Zaslow (1996) described how such pairs arise geometrically as torus fibra- tions with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich (1994) conjectured that a complex invariant on X (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of Y (the Fukaya category). This is known as homological mirror symmetry. In this project, we first use the construction of SYZ mirrors for hypersurfaces in abelian varieties following Abouzaid-Auroux-Katzarkov, in order to obtain X and Y as manifolds. -
D-Branes, RR-Fields and Duality on Noncommutative Manifolds
D-BRANES, RR-FIELDS AND DUALITY ON NONCOMMUTATIVE MANIFOLDS JACEK BRODZKI, VARGHESE MATHAI, JONATHAN ROSENBERG, AND RICHARD J. SZABO Abstract. We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincar´eduality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant K-theory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams. Contents Introduction 2 Acknowledgments 3 1. D-Branes and Ramond-Ramond Charges 4 1.1. Flat D-Branes 4 1.2. Ramond-Ramond Fields 7 1.3. Noncommutative D-Branes 9 1.4. Twisted D-Branes 10 2. Poincar´eDuality 13 2.1. Exterior Products in K-Theory 13 2.2. KK-Theory 14 2.3. Strong Poincar´eDuality 16 2.4. Duality Groups 18 arXiv:hep-th/0607020v3 26 Jun 2007 2.5. Spectral Triples 19 2.6. Twisted Group Algebra Completions of Surface Groups 21 2.7. Other Notions of Poincar´eDuality 22 3. KK-Equivalence 23 3.1. Strong KK-Equivalence 24 3.2. Other Notions of KK-Equivalence 25 3.3. Universal Coefficient Theorem 26 3.4. Deformations 27 3.5. Homotopy Equivalence 27 4. Cyclic Theory 27 4.1. Formal Properties of Cyclic Homology Theories 27 4.2. Local Cyclic Theory 29 5. -
Combinatorial Topology and Applications to Quantum Field Theory
Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Vivek Shende, Chair Professor Ian Agol Professor Constantin Teleman Professor Joel Moore Fall 2018 Abstract Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren Doctor of Philosophy in Mathematics University of California, Berkeley Professor Vivek Shende, Chair Topology has become increasingly important in the study of many-body quantum mechanics, in both high energy and condensed matter applications. While the importance of smooth topology has long been appreciated in this context, especially with the rise of index theory, torsion phenomena and dis- crete group symmetries are relatively new directions. In this thesis, I collect some mathematical results and conjectures that I have encountered in the exploration of these new topics. I also give an introduction to some quantum field theory topics I hope will be accessible to topologists. 1 To my loving parents, kind friends, and patient teachers. i Contents I Discrete Topology Toolbox1 1 Basics4 1.1 Discrete Spaces..........................4 1.1.1 Cellular Maps and Cellular Approximation.......6 1.1.2 Triangulations and Barycentric Subdivision......6 1.1.3 PL-Manifolds and Combinatorial Duality........8 1.1.4 Discrete Morse Flows...................9 1.2 Chains, Cycles, Cochains, Cocycles............... 13 1.2.1 Chains, Cycles, and Homology.............. 13 1.2.2 Pushforward of Chains.................. 15 1.2.3 Cochains, Cocycles, and Cohomology......... -
The Decomposition Theorem, Perverse Sheaves and the Topology Of
The decomposition theorem, perverse sheaves and the topology of algebraic maps Mark Andrea A. de Cataldo and Luca Migliorini∗ Abstract We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples. Contents 1 Overview 3 1.1 The topology of complex projective manifolds: Lefschetz and Hodge theorems 4 1.2 Families of smooth projective varieties . ........ 5 1.3 Singular algebraic varieties . ..... 7 1.4 Decomposition and hard Lefschetz in intersection cohomology . 8 1.5 Crash course on sheaves and derived categories . ........ 9 1.6 Decomposition, semisimplicity and relative hard Lefschetz theorems . 13 1.7 InvariantCycletheorems . 15 1.8 Afewexamples.................................. 16 1.9 The decomposition theorem and mixed Hodge structures . ......... 17 1.10 Historicalandotherremarks . 18 arXiv:0712.0349v2 [math.AG] 16 Apr 2009 2 Perverse sheaves 20 2.1 Intersection cohomology . 21 2.2 Examples of intersection cohomology . ...... 22 2.3 Definition and first properties of perverse sheaves . .......... 24 2.4 Theperversefiltration . .. .. .. .. .. .. .. 28 2.5 Perversecohomology .............................. 28 2.6 t-exactness and the Lefschetz hyperplane theorem . ...... 30 2.7 Intermediateextensions . 31 ∗Partially supported by GNSAGA and PRIN 2007 project “Spazi di moduli e teoria di Lie” 1 3 Three approaches to the decomposition theorem 33 3.1 The proof of Beilinson, Bernstein, Deligne and Gabber . -
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. -
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 . -
The Top Mathematics Award
Fields told me and which I later verified in Sweden, namely, that Nobel hated the mathematician Mittag- Leffler and that mathematics would not be one of the do- mains in which the Nobel prizes would The Top Mathematics be available." Award Whatever the reason, Nobel had lit- tle esteem for mathematics. He was Florin Diacuy a practical man who ignored basic re- search. He never understood its impor- tance and long term consequences. But Fields did, and he meant to do his best John Charles Fields to promote it. Fields was born in Hamilton, Ontario in 1863. At the age of 21, he graduated from the University of Toronto Fields Medal with a B.A. in mathematics. Three years later, he fin- ished his Ph.D. at Johns Hopkins University and was then There is no Nobel Prize for mathematics. Its top award, appointed professor at Allegheny College in Pennsylvania, the Fields Medal, bears the name of a Canadian. where he taught from 1889 to 1892. But soon his dream In 1896, the Swedish inventor Al- of pursuing research faded away. North America was not fred Nobel died rich and famous. His ready to fund novel ideas in science. Then, an opportunity will provided for the establishment of to leave for Europe arose. a prize fund. Starting in 1901 the For the next 10 years, Fields studied in Paris and Berlin annual interest was awarded yearly with some of the best mathematicians of his time. Af- for the most important contributions ter feeling accomplished, he returned home|his country to physics, chemistry, physiology or needed him. -
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. -
2003 Jean-Pierre Serre: an Overview of His Work
2003 Jean-Pierre Serre Jean-Pierre Serre: Mon premier demi-siècle au Collège de France Jean-Pierre Serre: My First Fifty Years at the Collège de France Marc Kirsch Ce chapitre est une interview par Marc Kirsch. Publié précédemment dans Lettre du Collège de France,no 18 (déc. 2006). Reproduit avec autorisation. This chapter is an interview by Marc Kirsch. Previously published in Lettre du Collège de France, no. 18 (déc. 2006). Reprinted with permission. M. Kirsch () Collège de France, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France e-mail: [email protected] H. Holden, R. Piene (eds.), The Abel Prize, 15 DOI 10.1007/978-3-642-01373-7_3, © Springer-Verlag Berlin Heidelberg 2010 16 Jean-Pierre Serre: Mon premier demi-siècle au Collège de France Jean-Pierre Serre, Professeur au Collège de France, titulaire de la chaire d’Algèbre et Géométrie de 1956 à 1994. Vous avez enseigné au Collège de France de 1956 à 1994, dans la chaire d’Algèbre et Géométrie. Quel souvenir en gardez-vous? J’ai occupé cette chaire pendant 38 ans. C’est une longue période, mais il y a des précédents: si l’on en croit l’Annuaire du Collège de France, au XIXe siècle, la chaire de physique n’a été occupée que par deux professeurs: l’un est resté 60 ans, l’autre 40. Il est vrai qu’il n’y avait pas de retraite à cette époque et que les pro- fesseurs avaient des suppléants (auxquels ils versaient une partie de leur salaire). Quant à mon enseignement, voici ce que j’en disais dans une interview de 19861: “Enseigner au Collège est un privilège merveilleux et redoutable. -
Presentation of the Austrian Mathematical Society - E-Mail: [email protected] La Rochelle University Lasie, Avenue Michel Crépeau B
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Features S E European A Problem for the 21st/22nd Century M M Mathematical Euler, Stirling and Wallis E S Society History Grothendieck: The Myth of a Break December 2019 Issue 114 Society ISSN 1027-488X The Austrian Mathematical Society Yerevan, venue of the EMS Executive Committee Meeting New 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. Todd Fisher (Brigham Young University, Provo, USA) and Boris Hasselblatt (Tufts University, Medford, USA) Hyperbolic Flows (Zürich Lectures in Advanced Mathematics) ISBN 978-3-03719-200-9. 2019. 737 pages. Softcover. 17 x 24 cm. 78.00 Euro The origins of dynamical systems trace back to flows and differential equations, and this is a modern text and reference on dynamical systems in which continuous-time dynamics is primary. It addresses needs unmet by modern books on dynamical systems, which largely focus on discrete time. Students have lacked a useful introduction to flows, and researchers have difficulty finding references to cite for core results in the theory of flows. Even when these are known substantial diligence and consulta- tion with experts is often needed to find them. This book presents the theory of flows from the topological, smooth, and measurable points of view. The first part introduces the general topological and ergodic theory of flows, and the second part presents the core theory of hyperbolic flows as well as a range of recent developments.