Index Theory For Locally Compact Noncommutative Geometries
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Index Theory For Locally Compact Noncommutative Geometries
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Author : A. L. Carey
language : en
Publisher: American Mathematical Soc.
Release Date : 2014-08-12
Index Theory For Locally Compact Noncommutative Geometries written by A. L. Carey and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-08-12 with Mathematics categories.
Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, the authors prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and the authors illustrate this point with two examples in the text. In order to understand what is new in their approach in the commutative setting the authors prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds their index formula appears to be completely new.
Advances In Noncommutative Geometry
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Author : Ali Chamseddine
language : en
Publisher: Springer Nature
Release Date : 2020-01-13
Advances In Noncommutative Geometry written by Ali Chamseddine and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-01-13 with Mathematics categories.
This authoritative volume in honor of Alain Connes, the foremost architect of Noncommutative Geometry, presents the state-of-the art in the subject. The book features an amalgam of invited survey and research papers that will no doubt be accessed, read, and referred to, for several decades to come. The pertinence and potency of new concepts and methods are concretely illustrated in each contribution. Much of the content is a direct outgrowth of the Noncommutative Geometry conference, held March 23–April 7, 2017, in Shanghai, China. The conference covered the latest research and future areas of potential exploration surrounding topology and physics, number theory, as well as index theory and its ramifications in geometry.
Noncommutative Geometry
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Author : Alain Connes
language : en
Publisher: Academic Press
Release Date : 1995-01-17
Noncommutative Geometry written by Alain Connes and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-01-17 with Mathematics categories.
This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields. Key Features * First full treatment of the subject and its applications * Written by the pioneer of this field * Broad applications in mathematics * Of interest across most fields * Ideal as an introduction and survey * Examples treated include: @subbul* the space of Penrose tilings * the space of leaves of a foliation * the space of irreducible unitary representations of a discrete group * the phase space in quantum mechanics * the Brillouin zone in the quantum Hall effect * A model of space time
Topics In Noncommutative Geometry
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Author : Guillermo Cortiñas
language : en
Publisher: American Mathematical Soc.
Release Date : 2012
Topics In Noncommutative Geometry written by Guillermo Cortiñas and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
Luis Santalo Winter Schools are organized yearly by the Mathematics Department and the Santalo Mathematical Research Institute of the School of Exact and Natural Sciences of the University of Buenos Aires (FCEN). This volume contains the proceedings of the third Luis Santalo Winter School which was devoted to noncommutative geometry and held at FCEN July 26-August 6, 2010. Topics in this volume concern noncommutative geometry in a broad sense, encompassing various mathematical and physical theories that incorporate geometric ideas to the study of noncommutative phenomena. It explores connections with several areas including algebra, analysis, geometry, topology and mathematical physics. Bursztyn and Waldmann discuss the classification of star products of Poisson structures up to Morita equivalence. Tsygan explains the connections between Kontsevich's formality theorem, noncommutative calculus, operads and index theory. Hoefel presents a concrete elementary construction in operad theory. Meyer introduces the subject of $\mathrm{C}^*$-algebraic crossed products. Rosenberg introduces Kasparov's $KK$-theory and noncommutative tori and includes a discussion of the Baum-Connes conjecture for $K$-theory of crossed products, among other topics. Lafont, Ortiz, and Sanchez-Garcia carry out a concrete computation in connection with the Baum-Connes conjecture. Zuk presents some remarkable groups produced by finite automata. Mesland discusses spectral triples and the Kasparov product in $KK$-theory. Trinchero explores the connections between Connes' noncommutative geometry and quantum field theory. Karoubi demonstrates a construction of twisted $K$-theory by means of twisted bundles. Tabuada surveys the theory of noncommutative motives.
Noncommutative Geometry And Physics
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Author : Alan L. Carey
language : en
Publisher: European Mathematical Society
Release Date : 2011
Noncommutative Geometry And Physics written by Alan L. Carey and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Geometry, Algebraic categories.
This collection of expository articles grew out of the workshop ``Number Theory and Physics'' held in March 2009 at The Erwin Schrodinger International Institute for Mathematical Physics, Vienna. The common theme of the articles is the influence of ideas from noncommutative geometry (NCG) on subjects ranging from number theory to Lie algebras, index theory, and mathematical physics. Matilde Marcolli's article gives a survey of relevant aspects of NCG in number theory, building on an introduction to motives for beginners by Jorge Plazas and Sujatha Ramdorai. A mildly unconventional view of index theory, from the viewpoint of NCG, is described in the article by Alan Carey, John Phillips, and Adam Rennie. As developed by Alain Connes and Dirk Kreimer, NCG also provides insight into novel algebraic structures underlying many analytic aspects of quantum field theory. Dominique Manchon's article on pre-Lie algebras fits into this developing research area. This interplay of algebraic and analytic techniques also appears in the articles by Christoph Bergbauer, who introduces renormalization theory and Feynman diagram methods, and Sylvie Paycha, who focuses on relations between renormalization and zeta function techniques.
Spectral Action In Noncommutative Geometry
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Author : Michał Eckstein
language : en
Publisher: Springer
Release Date : 2018-12-18
Spectral Action In Noncommutative Geometry written by Michał Eckstein and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-12-18 with Science categories.
What is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry à la Connes, deliberately unveiling the answers to these questions. After a brief preface flashing the panorama of the spectral approach, a concise primer on spectral triples is given. Chapter 2 is designed to serve as a toolkit for computations. The third chapter offers an in-depth view into the subtle links between the asymptotic expansions of traces of heat operators and meromorphic extensions of the associated spectral zeta functions. Chapter 4 studies the behaviour of the spectral action under fluctuations by gauge potentials. A subjective list of open problems in the field is spelled out in the fifth Chapter. The book concludes with an appendix including some auxiliary tools from geometry and analysis, along with examples of spectral geometries. The book serves both as a compendium for researchers in the domain of noncommutative geometry and an invitation to mathematical physicists looking for new concepts.
Elliptic Theory And Noncommutative Geometry
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Author : Vladimir E. Nazaykinskiy
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-06-30
Elliptic Theory And Noncommutative Geometry written by Vladimir E. Nazaykinskiy and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-06-30 with Mathematics categories.
This comprehensive yet concise book deals with nonlocal elliptic differential operators. These are operators whose coefficients involve shifts generated by diffeomorphisms of the manifold on which the operators are defined. This is the first book featuring a consistent application of methods of noncommutative geometry to the index problem in the theory of nonlocal elliptic operators. To make the book self-contained, the authors have included necessary geometric material.
A Computational Non Commutative Geometry Program For Disordered Topological Insulators
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Author : Emil Prodan
language : en
Publisher: Springer
Release Date : 2017-03-17
A Computational Non Commutative Geometry Program For Disordered Topological Insulators written by Emil Prodan and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-17 with Science categories.
This work presents a computational program based on the principles of non-commutative geometry and showcases several applications to topological insulators. Noncommutative geometry has been originally proposed by Jean Bellissard as a theoretical framework for the investigation of homogeneous condensed matter systems. Recently, this approach has been successfully applied to topological insulators, where it facilitated many rigorous results concerning the stability of the topological invariants against disorder.In the first part of the book the notion of a homogeneous material is introduced and the class of disordered crystals defined together with the classification table, which conjectures all topological phases from this class. The manuscript continues with a discussion of electrons’ dynamics in disordered crystals and the theory of topological invariants in the presence of strong disorder is briefly reviewed. It is shown how all this can be captured in the language of noncommutative geometry using the concept of non-commutative Brillouin torus, and a list of known formulas for various physical response functions is presented. In the second part, auxiliary algebras are introduced and a canonical finite-volume approximation of the non-commutative Brillouin torus is developed. Explicit numerical algorithms for computing generic correlation functions are discussed. In the third part upper bounds on the numerical errors are derived and it is proved that the canonical-finite volume approximation converges extremely fast to the thermodynamic limit. Convergence tests and various applications concludes the presentation.The book is intended for graduate students and researchers in numerical and mathematical physics.
Index Theory Beyond The Fredholm Case
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Author : Alan Carey
language : en
Publisher: Springer Nature
Release Date : 2022-11-30
Index Theory Beyond The Fredholm Case written by Alan Carey and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-11-30 with Mathematics categories.
This book is about extending index theory to some examples where non-Fredholm operators arise. It focuses on one aspect of the problem of what replaces the notion of spectral flow and the Fredholm index when the operators in question have zero in their essential spectrum. Most work in this topic stems from the so-called Witten index that is discussed at length here. The new direction described in these notes is the introduction of `spectral flow beyond the Fredholm case'. Creating a coherent picture of numerous investigations and scattered notions of the past 50 years, this work carefully introduces spectral flow, the Witten index and the spectral shift function and describes their relationship. After the introduction, Chapter 2 carefully reviews Double Operator Integrals, Chapter 3 describes the class of so-called p-relative trace class perturbations, followed by the construction of Krein's spectral shift function in Chapter 4. Chapter 5 reviews the analytic approach to spectral flow, culminating in Chapter 6 in the main abstract result of the book, namely the so-called principal trace formula. Chapter 7 completes the work with illustrations of the main results using explicit computations on two examples: the Dirac operator in Rd, and a differential operator on an interval. Throughout, attention is paid to the history of the subject and earlier references are provided accordingly. The book is aimed at experts in index theory as well as newcomers to the field.
The Connes Character Formula For Locally Compact Spectral Triples
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Author : F. A. Sukochev
language : en
Publisher:
Release Date : 2023
The Connes Character Formula For Locally Compact Spectral Triples written by F. A. Sukochev and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023 with Chern classes categories.
"A fundamental tool in noncommutative geometry is Connes's character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterization of manifolds. A non-compact space is modeled in noncommutative geometry by a non-unital spectral triple. The authors' aim is to establish Connes's character formula for non-unital spectral triples. This is significantly more difficult than in the unital case, and they achieve it with the use of recently developed double operator integration techniques. Previously, only partial extensions of Connes's character formula to the non-unital case were known. In the course of the proof, the authors establish two more results of importance in noncommutative geometry: an asymptotic for the heat semigroup of a non-unital spectral triple and the analyticity of the associated [Riemann zeta] function. The authors require certain assumptions on the underlying spectral triple and verify these assumptions in the case of spectral triples associated to arbitrary complete Riemannian manifolds and also in the case of Moyal planes"--American Mathematical Society bookstore website.