Topics in Non-Commutative Geometry

Topics in Non-Commutative Geometry
Title Topics in Non-Commutative Geometry PDF eBook
Author Y. Manin
Publisher Princeton University Press
Pages 173
Release 2014-07-14
Genre Mathematics
ISBN 1400862515

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There is a well-known correspondence between the objects of algebra and geometry: a space gives rise to a function algebra; a vector bundle over the space corresponds to a projective module over this algebra; cohomology can be read off the de Rham complex; and so on. In this book Yuri Manin addresses a variety of instances in which the application of commutative algebra cannot be used to describe geometric objects, emphasizing the recent upsurge of activity in studying noncommutative rings as if they were function rings on "noncommutative spaces." Manin begins by summarizing and giving examples of some of the ideas that led to the new concepts of noncommutative geometry, such as Connes' noncommutative de Rham complex, supergeometry, and quantum groups. He then discusses supersymmetric algebraic curves that arose in connection with superstring theory; examines superhomogeneous spaces, their Schubert cells, and superanalogues of Weyl groups; and provides an introduction to quantum groups. This book is intended for mathematicians and physicists with some background in Lie groups and complex geometry. Originally published in 1991. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Noncommutative Geometry and Number Theory

Noncommutative Geometry and Number Theory
Title Noncommutative Geometry and Number Theory PDF eBook
Author Caterina Consani
Publisher Springer Science & Business Media
Pages 374
Release 2007-12-18
Genre Mathematics
ISBN 3834803529

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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.

Noncommutative Geometry

Noncommutative Geometry
Title Noncommutative Geometry PDF eBook
Author Alain Connes
Publisher Springer
Pages 364
Release 2003-12-15
Genre Mathematics
ISBN 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.

Noncommutative Geometry, Arithmetic, and Related Topics

Noncommutative Geometry, Arithmetic, and Related Topics
Title Noncommutative Geometry, Arithmetic, and Related Topics PDF eBook
Author Caterina Consani
Publisher JHU Press
Pages 324
Release 2011
Genre Mathematics
ISBN 1421403528

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Mathematics Institute, these essays collectively provide mathematicians and physicists with a comprehensive resource on the topic.

Advances in Noncommutative Geometry

Advances in Noncommutative Geometry
Title Advances in Noncommutative Geometry PDF eBook
Author Ali Chamseddine
Publisher Springer Nature
Pages 753
Release 2020-01-13
Genre Mathematics
ISBN 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.

Elements of Noncommutative Geometry

Elements of Noncommutative Geometry
Title Elements of Noncommutative Geometry PDF eBook
Author Jose M. Gracia-Bondia
Publisher Springer Science & Business Media
Pages 692
Release 2013-11-27
Genre Mathematics
ISBN 1461200059

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Noncommutative Geometry and Particle Physics

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

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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.