Local And Analytic Cyclic Homology

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Local and Analytic Cyclic Homology

Ralf Meyer
Local and Analytic Cyclic Homology Book PDF

Author : Ralf Meyer
Publisher : European Mathematical Society
Page : 376 pages
File Size : 51,6 Mb
Release : 2007
Category : Mathematics
ISBN : 3037190396

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Local and Analytic Cyclic Homology by Ralf Meyer Pdf

Periodic cyclic homology is a homology theory for non-commutative algebras that plays a similar role in non-commutative geometry as de Rham cohomology for smooth manifolds. While it produces good results for algebras of smooth or polynomial functions, it fails for bigger algebras such as most Banach algebras or C*-algebras. Analytic and local cyclic homology are variants of periodic cyclic homology that work better for such algebras. In this book, the author develops and compares these theories, emphasizing their homological properties. This includes the excision theorem, invariance under passage to certain dense subalgebras, a Universal Coefficient Theorem that relates them to $K$-theory, and the Chern-Connes character for $K$-theory and $K$-homology. The cyclic homology theories studied in this text require a good deal of functional analysis in bornological vector spaces, which is supplied in the first chapters. The focal points here are the relationship with inductive systems and the functional calculus in non-commutative bornological algebras. Some chapters are more elementary and independent of the rest of the book and will be of interest to researchers and students working on functional analysis and its applications.

Cyclic Homology in Non-Commutative Geometry

Joachim Cuntz,Georges Skandalis,Boris Tsygan
Cyclic Homology in Non Commutative Geometry Book PDF

Author : Joachim Cuntz,Georges Skandalis,Boris Tsygan
Publisher : Springer Science & Business Media
Page : 160 pages
File Size : 51,6 Mb
Release : 2003-11-17
Category : Mathematics
ISBN : 3540404694

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Cyclic Homology in Non-Commutative Geometry by Joachim Cuntz,Georges Skandalis,Boris Tsygan Pdf

Contributions by three authors treat aspects of noncommutative geometry that are related to cyclic homology. The authors give rather complete accounts of cyclic theory from different points of view. The connections between (bivariant) K-theory and cyclic theory via generalized Chern-characters are discussed in detail. Cyclic theory is the natural setting for a variety of general abstract index theorems. A survey of such index theorems is given and the concepts and ideas involved in these theorems are explained.

Cyclic Cohomology at 40: Achievements and Future Prospects

A. Connes,C. Consani,B. I. Dundas,M. Khalkhali,H. Moscovici
Cyclic Cohomology at 40 Achievements and Future Prospects Book PDF

Author : A. Connes,C. Consani,B. I. Dundas,M. Khalkhali,H. Moscovici
Publisher : American Mathematical Society
Page : 592 pages
File Size : 45,5 Mb
Release : 2023-02-23
Category : Mathematics
ISBN : 9781470469771

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Cyclic Cohomology at 40: Achievements and Future Prospects by A. Connes,C. Consani,B. I. Dundas,M. Khalkhali,H. Moscovici Pdf

This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada. Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory. The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher categorical algebra, Hopf algebra symmetries, de Rham-Witt complex, quantum physics, etc., in which cyclic homology plays the role of a unifying theme. The researcher will find frontier research articles in which the cyclic theory provides a computational tool of great relevance. In particular, in analysis cyclic cohomology index formulas capture the higher invariants of manifolds, where the group symmetries are extended to Hopf algebra actions, and where Lie algebra cohomology is greatly extended to the cyclic cohomology of Hopf algebras which becomes the natural receptacle for characteristic classes. In algebraic topology the cyclotomic structure obtained using the cyclic subgroups of the circle action on topological Hochschild homology gives rise to remarkably significant arithmetic structures intimately related to crystalline cohomology through the de Rham-Witt complex, Fontaine's theory and the Fargues-Fontaine curve.

Perspectives on Noncommutative Geometry

Masoud Khalkhali
Perspectives on Noncommutative Geometry Book PDF

Author : Masoud Khalkhali
Publisher : American Mathematical Soc.
Page : 176 pages
File Size : 44,7 Mb
Release : 2011
Category : Algebra, Homological
ISBN : 9780821848494

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Perspectives on Noncommutative Geometry by Masoud Khalkhali Pdf

This volume represents the proceedings of the Noncommutative Geometry Workshop that was held as part of the thematic program on operator algebras at the Fields Institute in May 2008. Pioneered by Alain Connes starting in the late 1970s, noncommutative geometry was originally inspired by global analysis, topology, operator algebras, and quantum physics. Its main applications were to settle some long-standing conjectures, such as the Novikov conjecture and the Baum-Connes conjecture. Next came the impact of spectral geometry and the way the spectrum of a geometric operator, like the Laplacian, holds information about the geometry and topology of a manifold, as in the celebrated Weyl law. This has now been vastly generalized through Connes' notion of spectral triples. Finally, recent years have witnessed the impact of number theory, algebraic geometry and the theory of motives, and quantum field theory on noncommutative geometry. Almost all of these aspects are touched upon with new results in the papers of this volume. This book is intended for graduate students and researchers in both mathematics and theoretical physics who are interested in noncommutative geometry and its applications.

Efficient Numerical Methods for Non-local Operators

Steffen Börm
Efficient Numerical Methods for Non local Operators Book PDF

Author : Steffen Börm
Publisher : European Mathematical Society
Page : 452 pages
File Size : 51,9 Mb
Release : 2010
Category : Matrices
ISBN : 3037190914

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Efficient Numerical Methods for Non-local Operators by Steffen Börm Pdf

Hierarchical matrices present an efficient way of treating dense matrices that arise in the context of integral equations, elliptic partial differential equations, and control theory. While a dense $n\times n$ matrix in standard representation requires $n^2$ units of storage, a hierarchical matrix can approximate the matrix in a compact representation requiring only $O(n k \log n)$ units of storage, where $k$ is a parameter controlling the accuracy. Hierarchical matrices have been successfully applied to approximate matrices arising in the context of boundary integral methods, to construct preconditioners for partial differential equations, to evaluate matrix functions, and to solve matrix equations used in control theory. $\mathcal{H}^2$-matrices offer a refinement of hierarchical matrices: Using a multilevel representation of submatrices, the efficiency can be significantly improved, particularly for large problems. This book gives an introduction to the basic concepts and presents a general framework that can be used to analyze the complexity and accuracy of $\mathcal{H}^2$-matrix techniques. Starting from basic ideas of numerical linear algebra and numerical analysis, the theory is developed in a straightforward and systematic way, accessible to advanced students and researchers in numerical mathematics and scientific computing. Special techniques are required only in isolated sections, e.g., for certain classes of model problems.

Functional Equations and Characterization Problems on Locally Compact Abelian Groups

Gennadiĭ Mikhaĭlovich Felʹdman
Functional Equations and Characterization Problems on Locally Compact Abelian Groups Book PDF

Author : Gennadiĭ Mikhaĭlovich Felʹdman
Publisher : European Mathematical Society
Page : 272 pages
File Size : 51,9 Mb
Release : 2008
Category : Abelian groups
ISBN : 3037190450

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Functional Equations and Characterization Problems on Locally Compact Abelian Groups by Gennadiĭ Mikhaĭlovich Felʹdman Pdf

This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables. In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group $X$. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac-Bernstein, Skitovich-Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive definite functions defined on the character group of $X$. Group analogs of the Cramer and Marcinkiewicz theorems are also studied. The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory.

Cyclic Homology

Jean-Louis Loday
Cyclic Homology Book PDF

Author : Jean-Louis Loday
Publisher : Springer Science & Business Media
Page : 525 pages
File Size : 51,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662113899

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Cyclic Homology by Jean-Louis Loday Pdf

From the reviews: "This is a very interesting book containing material for a comprehensive study of the cyclid homological theory of algebras, cyclic sets and S1-spaces. Lie algebras and algebraic K-theory and an introduction to Connes'work and recent results on the Novikov conjecture. The book requires a knowledge of homological algebra and Lie algebra theory as well as basic technics coming from algebraic topology. The bibliographic comments at the end of each chapter offer good suggestions for further reading and research. The book can be strongly recommended to anybody interested in noncommutative geometry, contemporary algebraic topology and related topics." European Mathematical Society Newsletter In this second edition the authors have added a chapter 13 on MacLane (co)homology.

New Spaces in Physics

Mathieu Anel,Gabriel Catren
New Spaces in Physics Book PDF

Author : Mathieu Anel,Gabriel Catren
Publisher : Cambridge University Press
Page : 437 pages
File Size : 40,6 Mb
Release : 2021-04
Category : Mathematics
ISBN : 9781108490627

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New Spaces in Physics by Mathieu Anel,Gabriel Catren Pdf

In this graduate-level book, leading researchers explore various new notions of 'space' in mathematical physics.

Basic Noncommutative Geometry

Masoud Khalkhali
Basic Noncommutative Geometry Book PDF

Author : Masoud Khalkhali
Publisher : European Mathematical Society
Page : 244 pages
File Size : 48,5 Mb
Release : 2009
Category : Mathematics
ISBN : 3037190612

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Basic Noncommutative Geometry by Masoud Khalkhali Pdf

"Basic Noncommutative Geometry provides an introduction to noncommutative geometry and some of its applications. The book can be used either as a textbook for a graduate course on the subject or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes-Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well."--Publisher's description.

Nonlinear Potential Theory on Metric Spaces

Anders Björn,Jana Björn
Nonlinear Potential Theory on Metric Spaces Book PDF

Author : Anders Björn,Jana Björn
Publisher : European Mathematical Society
Page : 422 pages
File Size : 48,8 Mb
Release : 2011
Category : Harmonic functions
ISBN : 303719099X

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Nonlinear Potential Theory on Metric Spaces by Anders Björn,Jana Björn Pdf

The $p$-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory, and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs, and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for interested readers and as a reference text for active researchers. The presentation is rather self contained, but it is assumed that readers know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces, and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references, and an extensive index is provided at the end of the book.

K-theory and Noncommutative Geometry

Guillermo Cortiñas
K theory and Noncommutative Geometry Book PDF

Author : Guillermo Cortiñas
Publisher : European Mathematical Society
Page : 460 pages
File Size : 43,6 Mb
Release : 2008
Category : K-theory
ISBN : 3037190604

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K-theory and Noncommutative Geometry by Guillermo Cortiñas Pdf

Since its inception 50 years ago, K-theory has been a tool for understanding a wide-ranging family of mathematical structures and their invariants: topological spaces, rings, algebraic varieties and operator algebras are the dominant examples. The invariants range from characteristic classes in cohomology, determinants of matrices, Chow groups of varieties, as well as traces and indices of elliptic operators. Thus K-theory is notable for its connections with other branches of mathematics. Noncommutative geometry develops tools which allow one to think of noncommutative algebras in the same footing as commutative ones: as algebras of functions on (noncommutative) spaces. The algebras in question come from problems in various areas of mathematics and mathematical physics; typical examples include algebras of pseudodifferential operators, group algebras, and other algebras arising from quantum field theory. To study noncommutative geometric problems one considers invariants of the relevant noncommutative algebras. These invariants include algebraic and topological K-theory, and also cyclic homology, discovered independently by Alain Connes and Boris Tsygan, which can be regarded both as a noncommutative version of de Rham cohomology and as an additive version of K-theory. There are primary and secondary Chern characters which pass from K-theory to cyclic homology. These characters are relevant both to noncommutative and commutative problems and have applications ranging from index theorems to the detection of singularities of commutative algebraic varieties. The contributions to this volume represent this range of connections between K-theory, noncommmutative geometry, and other branches of mathematics.

Cyclic Homology in Non-Commutative Geometry

Joachim Cuntz,Georges Skandalis,Boris Tsygan
Cyclic Homology in Non Commutative Geometry Book PDF

Author : Joachim Cuntz,Georges Skandalis,Boris Tsygan
Publisher : Springer
Page : 0 pages
File Size : 48,6 Mb
Release : 2011-01-23
Category : Mathematics
ISBN : 3642073379

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Cyclic Homology in Non-Commutative Geometry by Joachim Cuntz,Georges Skandalis,Boris Tsygan Pdf

Contributions by three authors treat aspects of noncommutative geometry that are related to cyclic homology. The authors give rather complete accounts of cyclic theory from different points of view. The connections between (bivariant) K-theory and cyclic theory via generalized Chern-characters are discussed in detail. Cyclic theory is the natural setting for a variety of general abstract index theorems. A survey of such index theorems is given and the concepts and ideas involved in these theorems are explained.

Topological and Bivariant K-Theory

Joachim Cuntz,Jonathan M. Rosenberg
Topological and Bivariant K Theory Book PDF

Author : Joachim Cuntz,Jonathan M. Rosenberg
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 49,7 Mb
Release : 2007-10-04
Category : Mathematics
ISBN : 9783764383992

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Topological and Bivariant K-Theory by Joachim Cuntz,Jonathan M. Rosenberg Pdf

Topological K-theory is one of the most important invariants for noncommutative algebras. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory. This book describes a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, it details other approaches to bivariant K-theories for operator algebras. The book studies a number of applications, including K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem.

Triangulated Categories

Thorsten Holm,Peter Jørgensen,Raphaël Rouquier
Triangulated Categories Book PDF

Author : Thorsten Holm,Peter Jørgensen,Raphaël Rouquier
Publisher : Cambridge University Press
Page : 473 pages
File Size : 48,7 Mb
Release : 2010-06-24
Category : Mathematics
ISBN : 9781139488884

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Triangulated Categories by Thorsten Holm,Peter Jørgensen,Raphaël Rouquier Pdf

A 2010 collection of survey articles by leading experts covering fundamental aspects of triangulated categories, as well as applications in algebraic geometry, representation theory, commutative algebra, microlocal analysis and algebraic topology. This is a valuable reference for experts and a useful introduction for graduate students entering the field.

Topics in Algebraic and Topological K-Theory

Paul Frank Baum,Ralf Meyer,Rubén Sánchez-García,Marco Schlichting,Bertrand Toën
Topics in Algebraic and Topological K Theory Book PDF

Author : Paul Frank Baum,Ralf Meyer,Rubén Sánchez-García,Marco Schlichting,Bertrand Toën
Publisher : Springer Science & Business Media
Page : 322 pages
File Size : 40,5 Mb
Release : 2010-11-05
Category : Mathematics
ISBN : 9783642157073

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Topics in Algebraic and Topological K-Theory by Paul Frank Baum,Ralf Meyer,Rubén Sánchez-García,Marco Schlichting,Bertrand Toën Pdf

This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.