Cyclic Homology in Non-Commutative Geometry

Cyclic Homology in Non-Commutative Geometry

Author: Joachim Cuntz

Publisher: Springer Science & Business Media

Published: 2003-11-17

Total Pages: 160

ISBN-13: 9783540404699

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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.


Noncommutative Geometry

Noncommutative Geometry

Author: Alain Connes

Publisher: Springer

Published: 2003-12-15

Total Pages: 364

ISBN-13: 3540397027

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Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.


Cyclic Homology in Non-Commutative Geometry

Cyclic Homology in Non-Commutative Geometry

Author: Joachim Cuntz

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 147

ISBN-13: 3662064448

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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 Homology

Cyclic Homology

Author: Jean-Louis Loday

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 467

ISBN-13: 3662217392

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This book is a comprehensive study of cyclic homology theory together with its relationship with Hochschild homology, de Rham cohomology, S1 equivariant homology, the Chern character, Lie algebra homology, algebraic K-theory and non-commutative differential geometry. Though conceived as a basic reference on the subject, many parts of this book are accessible to graduate students.


From Differential Geometry to Non-commutative Geometry and Topology

From Differential Geometry to Non-commutative Geometry and Topology

Author: Neculai S. Teleman

Publisher: Springer Nature

Published: 2019-11-10

Total Pages: 406

ISBN-13: 3030284336

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This book aims to provide a friendly introduction to non-commutative geometry. It studies index theory from a classical differential geometry perspective up to the point where classical differential geometry methods become insufficient. It then presents non-commutative geometry as a natural continuation of classical differential geometry. It thereby aims to provide a natural link between classical differential geometry and non-commutative geometry. The book shows that the index formula is a topological statement, and ends with non-commutative topology.


Advances in Noncommutative Geometry

Advances in Noncommutative Geometry

Author: Ali Chamseddine

Publisher: Springer Nature

Published: 2020-01-13

Total Pages: 753

ISBN-13: 3030295974

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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.


Homological Mirror Symmetry

Homological Mirror Symmetry

Author: Anton Kapustin

Publisher: Springer Science & Business Media

Published: 2009

Total Pages: 281

ISBN-13: 3540680292

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An ideal reference on the mathematical aspects of quantum field theory, this volume provides a set of lectures and reviews that both introduce and representatively review the state-of-the art in the field from different perspectives.


Cyclic Homology in Non-Commutative Geometry

Cyclic Homology in Non-Commutative Geometry

Author: Joachim Cuntz

Publisher: Springer

Published: 2014-01-15

Total Pages: 156

ISBN-13: 9783662064450

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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.


Noncommutative Geometry, Quantum Fields and Motives

Noncommutative Geometry, Quantum Fields and Motives

Author: Alain Connes

Publisher: American Mathematical Soc.

Published: 2019-03-13

Total Pages: 810

ISBN-13: 1470450453

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The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.