Perspectives On Noncommutative Geometry

Author : Masoud Khalkhali
Publisher : American Mathematical Soc.
Page : 176 pages
File Size : 50,9 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.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 43,7 Mb
Release : 2012
Category : Algebra, Homological
ISBN : 1470417820

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Perspectives on Noncommutative Geometry by Anonim Pdf

Author : Neculai S. Teleman
Publisher : Springer Nature
Page : 398 pages
File Size : 47,8 Mb
Release : 2019-11-10
Category : Mathematics
ISBN : 9783030284336

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From Differential Geometry to Non-commutative Geometry and Topology by Neculai S. Teleman Pdf

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.

Author : Supriya Kar
Publisher : World Scientific Publishing Company Incorporated
Page : 250 pages
File Size : 43,8 Mb
Release : 2018-01-31
Category : Mathematics
ISBN : 9812380523

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Non-commutative Geometry by Supriya Kar Pdf

This book provides a systematic, comprehensive and up-to-date account of the recent developments in non-commutative geometry, at a pedagogical level. It does not go into the details of rigorous (advanced level) mathematical formulation of non-commutative geometry; rather, it restricts itself to the domain of strings and quantum fields.Since non-commutative geometry has recently aroused renewed interest in open string theory, the author motivates the text from the viewpoint of a string theory. He begins with an introduction to the subject, explaining what one means by non-commutative geometry and why it is relevant to study such geometry, and discussing its possible origin in a string theory.The book comprises nine chapters. Chapter 1 gives a sound mathematical ntroduction to non-commutative spacetime coordinates in classical and quantum physics. In Chapter 2, non-commutativity in a string theory is discussed at a pedagogic level. Chapter 3 deals with an aribitrary D-brane dynamics and Chapter 4 describes the non-commutative gauge theories on a D-brane. In Chapters 5-9, non-commutative quantum field theory (NCQFT) is addressed. In particular, Chapter 5 deals with the real scalar NCQFT, Chapter 6 with that of complex scalar field, Chapter 7 describes spontaneous symmetry breaking in scalar NCQFT, Chapter 8 deals with the U(1) Gauge theory and Chapter 9 with SU(n) Gauge theories.Students will find this book useful as a bridge between string and field theories. In addition, it will prove invaluable for interdisciplinary areas of study.

Author : Caterina Consani,Matilde Marcolli
Publisher : Springer Science & Business Media
Page : 372 pages
File Size : 40,6 Mb
Release : 2007-12-18
Category : Mathematics
ISBN : 9783834803528

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Noncommutative Geometry and Number Theory by Caterina Consani,Matilde Marcolli Pdf

In recent years, number theory and arithmetic geometry have been enriched by new techniques from noncommutative geometry, operator algebras, dynamical systems, and K-Theory. This volume collects and presents up-to-date research topics in arithmetic and noncommutative geometry and ideas from physics that point to possible new connections between the fields of number theory, algebraic geometry and noncommutative geometry. The articles collected in this volume present new noncommutative geometry perspectives on classical topics of number theory and arithmetic such as modular forms, class field theory, the theory of reductive p-adic groups, Shimura varieties, the local L-factors of arithmetic varieties. They also show how arithmetic appears naturally in noncommutative geometry and in physics, in the residues of Feynman graphs, in the properties of noncommutative tori, and in the quantum Hall effect.

Author : Matilde Marcolli,Deepak Parashar
Publisher : Springer Science & Business Media
Page : 240 pages
File Size : 46,9 Mb
Release : 2010-11-02
Category : Mathematics
ISBN : 9783834898319

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Quantum Groups and Noncommutative Spaces by Matilde Marcolli,Deepak Parashar Pdf

This book is aimed at presenting different methods and perspectives in the theory of Quantum Groups, bridging between the algebraic, representation theoretic, analytic, and differential-geometric approaches. It also covers recent developments in Noncommutative Geometry, which have close relations to quantization and quantum group symmetries. The volume collects surveys by experts which originate from an acitvity at the Max-Planck-Institute for Mathematics in Bonn.

Author : Giuseppe Dito,Motoko Kotani,Yoshiaki Maeda,Hitoshi Moriyoshi,Toshikazu Natsume,Satoshi Watamura
Publisher : World Scientific
Page : 536 pages
File Size : 55,6 Mb
Release : 2013-01-11
Category : Mathematics
ISBN : 9789814425025

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Noncommutative Geometry and Physics 3 by Giuseppe Dito,Motoko Kotani,Yoshiaki Maeda,Hitoshi Moriyoshi,Toshikazu Natsume,Satoshi Watamura Pdf

Noncommutative differential geometry is a novel approach to geometry, aimed in part at applications in physics. It was founded in the early eighties by the 1982 Fields Medalist Alain Connes on the basis of his fundamental works in operator algebras. It is now a very active branch of mathematics with actual and potential applications to a variety of domains in physics ranging from solid state to quantization of gravity. The strategy is to formulate usual differential geometry in a somewhat unusual manner, using in particular operator algebras and related concepts, so as to be able to plug in noncommutativity in a natural way. Algebraic tools such as K-theory and cyclic cohomology and homology play an important role in this field. It is an important topic both for mathematics and physics. Contents:K-Theory and D-Branes, Shonan:The Local Index Formula in Noncommutative Geometry Revisited (Alan L Carey, John Phillips, Adam Rennie and Fedor A Sukochev)Semi-Finite Noncommutative Geometry and Some Applications (Alan L Carey, John Phillips and Adam Rennie)Generalized Geometries in String Compactification Scenarios (Tetsuji Kimura)What Happen to Gauge Theories under Noncommutative Deformation? (Akifumi Sako)D-Branes and Bivariant K-Theory (Richard J Szabo)Two-Sided Bar Constructions for Partial Monoids and Applications to K-Homology Theory (Dai Tamaki)Twisting Segal's K-Homology Theory (Dai Tamaki)Spectrum of Non-Commutative Harmonic Oscillators and Residual Modular Forms (Kazufumi Kimoto and Masato Wakayama)Coarse Embeddings and Higher Index Problems for Expanders (Qin Wang)Deformation Quantization and Noncommutative Geometry, RIMS:Enriched Fell Bundles and Spaceoids (Paolo Bertozzini, Roberto Conti and Wicharn Lewkeeratiyutkul)Weyl Character Formula in KK-Theory (Jonathan Block and Nigel Higson)Recent Advances in the Study of the Equivariant Brauer Group (Peter Bouwknegt, Alan Carey and Rishni Ratnam)Entire Cyclic Cohomology of Noncommutative Manifolds (Katsutoshi Kawashima)Geometry of Quantum Projective Spaces (Francesco D'Andrea and Giovanni Landi)On Yang–Mills Theory for Quantum Heisenberg Manifolds (Hyun Ho Lee)Dilatational Equivalence Classes and Novikov–Shubin Type Capacities of Groups, and Random Walks (Shin-ichi Oguni)Deformation Quantization of Gauge Theory in ℝ4 and U(1) Instanton Problems (Yoshiaki Maeda and Akifumi Sako)Dualities in Field Theories and the Role of K-Theory (Jonathan Rosenberg)Dualities in Field Theories and the Role of K-Theory (Jonathan Rosenberg) Readership: Researchers and graduate students in Mathematical Physics and Applied Mathematics. Keywords:Noncommutative Geometry;Deformation Quantizations;D-Brane;K-Theory;T-Duality

Author : Jose M. Gracia-Bondia,Joseph C. Varilly,Hector Figueroa
Publisher : Springer Science & Business Media
Page : 692 pages
File Size : 54,5 Mb
Release : 2013-11-27
Category : Mathematics
ISBN : 9781461200055

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Elements of Noncommutative Geometry by Jose M. Gracia-Bondia,Joseph C. Varilly,Hector Figueroa Pdf

Author : Alain Connes,Matilde Marcolli
Publisher : American Mathematical Soc.
Page : 785 pages
File Size : 53,7 Mb
Release : 2019-03-13
Category : Electronic
ISBN : 9781470450458

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Noncommutative Geometry, Quantum Fields and Motives by Alain Connes,Matilde Marcolli Pdf

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.

Author : Walter D. van Suijlekom
Publisher : Springer
Page : 246 pages
File Size : 48,7 Mb
Release : 2014-07-21
Category : Science
ISBN : 9789401791625

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Noncommutative Geometry and Particle Physics by Walter D. van Suijlekom Pdf

This book provides an introduction to noncommutative geometry and presents a number of its recent applications to particle physics. It is intended for graduate students in mathematics/theoretical physics who are new to the field of noncommutative geometry, as well as for researchers in mathematics/theoretical physics with an interest in the physical applications of noncommutative geometry. In the first part, we introduce the main concepts and techniques by studying finite noncommutative spaces, providing a “light” approach to noncommutative geometry. We then proceed with the general framework by defining and analyzing noncommutative spin manifolds and deriving some main results on them, such as the local index formula. In the second part, we show how noncommutative spin manifolds naturally give rise to gauge theories, applying this principle to specific examples. We subsequently geometrically derive abelian and non-abelian Yang-Mills gauge theories, and eventually the full Standard Model of particle physics, and conclude by explaining how noncommutative geometry might indicate how to proceed beyond the Standard Model.

Author : Hossein Abbaspour,Matilde Marcolli,Thomas Tradler
Publisher : Springer Science & Business Media
Page : 173 pages
File Size : 41,8 Mb
Release : 2010-04-21
Category : Mathematics
ISBN : 9783834896803

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Deformation Spaces by Hossein Abbaspour,Matilde Marcolli,Thomas Tradler Pdf

The first instances of deformation theory were given by Kodaira and Spencer for complex structures and by Gerstenhaber for associative algebras. Since then, deformation theory has been applied as a useful tool in the study of many other mathematical structures, and even today it plays an important role in many developments of modern mathematics. This volume collects a few self-contained and peer-reviewed papers by experts which present up-to-date research topics in algebraic and motivic topology, quantum field theory, algebraic geometry, noncommutative geometry and the deformation theory of Poisson algebras. They originate from activities at the Max-Planck-Institute for Mathematics and the Hausdorff Center for Mathematics in Bonn.

Author : J. Madore
Publisher : Cambridge University Press
Page : 381 pages
File Size : 51,8 Mb
Release : 1999-06-24
Category : Mathematics
ISBN : 9780521659918

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An Introduction to Noncommutative Differential Geometry and Its Physical Applications by J. Madore Pdf

A thoroughly revised introduction to non-commutative geometry.

Author : Fred Van Oystaeyen
Publisher : CRC Press
Page : 168 pages
File Size : 53,8 Mb
Release : 2007-11-15
Category : Mathematics
ISBN : 1420060570

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Virtual Topology and Functor Geometry by Fred Van Oystaeyen Pdf

Intrinsically noncommutative spaces today are considered from the perspective of several branches of modern physics, including quantum gravity, string theory, and statistical physics. From this point of view, it is ideal to devise a concept of space and its geometry that is fundamentally noncommutative. Providing a clear introduction to noncommutative topology, Virtual Topology and Functor Geometry explores new aspects of these areas as well as more established facets of noncommutative algebra. Presenting the material in an easy, colloquial style to facilitate understanding, the book begins with an introduction to category theory, followed by a chapter on noncommutative spaces. This chapter examines noncommutative lattices, noncommutative opens, sheaf theory, the generalized Stone space, and Grothendieck topology. The author then studies Grothendieck categorical representations to formulate an abstract notion of "affine open". The final chapter proposes a dynamical version of topology and sheaf theory, providing at least one solution of the problem of sheafification independent of generalizations of topos theory. By presenting new ideas for the development of an intrinsically noncommutative geometry, this book fosters the further unification of different kinds of noncommutative geometry and the expression of observations that involve natural phenomena.

Author : Masoud Khalkhali
Publisher : European Mathematical Society
Page : 244 pages
File Size : 55,7 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.

Author : Igor V. Nikolaev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 403 pages
File Size : 40,7 Mb
Release : 2017-11-07
Category : Mathematics
ISBN : 9783110543483

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Noncommutative Geometry by Igor V. Nikolaev Pdf

This book covers the basics of noncommutative geometry (NCG) and its applications in topology, algebraic geometry, and number theory. The author takes up the practical side of NCG and its value for other areas of mathematics. A brief survey of the main parts of NCG with historical remarks, bibliography, and a list of exercises is included. The presentation is intended for graduate students and researchers with interests in NCG, but will also serve nonexperts in the field. Contents Part I: Basics Model examples Categories and functors C∗-algebras Part II: Noncommutative invariants Topology Algebraic geometry Number theory Part III: Brief survey of NCG Finite geometries Continuous geometries Connes geometries Index theory Jones polynomials Quantum groups Noncommutative algebraic geometry Trends in noncommutative geometry