Advances in Noncommutative Geometry

Advances in Noncommutative Geometry
Author: Ali Chamseddine
Publisher: Springer Nature
Total Pages: 753
Release: 2020-01-13
Genre: Mathematics
ISBN: 3030295974

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.

Spectral Action in Noncommutative Geometry

Spectral Action in Noncommutative Geometry
Author: Michał Eckstein
Publisher: Springer
Total Pages: 165
Release: 2018-12-18
Genre: Science
ISBN: 3319947885

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.

Noncommutative Geometry

Noncommutative Geometry
Author: Alain Connes
Publisher: Springer
Total Pages: 364
Release: 2003-12-15
Genre: Mathematics
ISBN: 3540397027

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.

Advances in Algebra and Geometry

Advances in Algebra and Geometry
Author: C. Musili
Publisher: Springer
Total Pages: 311
Release: 2003-01-01
Genre: Mathematics
ISBN: 9386279126

Contributed articles presented at the Conference, sponsored by National Science Foundation, USA [and others].

An Invitation To Noncommutative Geometry

An Invitation To Noncommutative Geometry
Author: Matilde Marcolli
Publisher: World Scientific
Total Pages: 515
Release: 2008-02-11
Genre: Science
ISBN: 9814475629

This is the first existing volume that collects lectures on this important and fast developing subject in mathematics. The lectures are given by leading experts in the field and the range of topics is kept as broad as possible by including both the algebraic and the differential aspects of noncommutative geometry as well as recent applications to theoretical physics and number theory.

Noncommutative Geometry

Noncommutative Geometry
Author: Igor V. Nikolaev
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 330
Release: 2017-11-07
Genre: Mathematics
ISBN: 3110543486

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

Noncommutative Geometry and Particle Physics

Noncommutative Geometry and Particle Physics
Author: Walter D. van Suijlekom
Publisher: Springer
Total Pages: 246
Release: 2014-07-21
Genre: Science
ISBN: 9401791627

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.