Finite Dimensional Approximations of Quantum Tori for the Quantum Gromov-Hausdorff Distance
Author: Frédéric Latrémolière
Publisher:
Published: 2004
Total Pages: 302
ISBN-13:
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Author: Frédéric Latrémolière
Publisher:
Published: 2004
Total Pages: 302
ISBN-13:
DOWNLOAD EBOOKAuthor: Pierre Martinetti
Publisher: American Mathematical Soc.
Published: 2016-10-26
Total Pages: 234
ISBN-13: 1470422972
DOWNLOAD EBOOKThe distance formula in noncommutative geometry was introduced by Connes at the end of the 1980s. It is a generalization of Riemannian geodesic distance that makes sense in a noncommutative setting, and provides an original tool to study the geometry of the space of states on an algebra. It also has an intriguing echo in physics, for it yields a metric interpretation for the Higgs field. In the 1990s, Rieffel noticed that this distance is a noncommutative version of the Wasserstein distance of order 1 in the theory of optimal transport. More exactly, this is a noncommutative generalization of Kantorovich dual formula of the Wasserstein distance. Connes distance thus offers an unexpected connection between an ancient mathematical problem and the most recent discovery in high energy physics. The meaning of this connection is far from clear. Yet, Rieffel's observation suggests that Connes distance may provide an interesting starting point for a theory of optimal transport in noncommutative geometry. This volume contains several review papers that will give the reader an extensive introduction to the metric aspect of noncommutative geometry and its possible interpretation as a Wasserstein distance on a quantum space, as well as several topic papers.
Author: Hanfeng Li
Publisher:
Published: 2002
Total Pages: 294
ISBN-13:
DOWNLOAD EBOOKAuthor: Azita Mayeli
Publisher: American Mathematical Soc.
Published: 2013-11-08
Total Pages: 218
ISBN-13: 0821894935
DOWNLOAD EBOOKThis volume contains the proceedings of the AMS Special Session on Wavelet and Frame Theoretic Methods in Harmonic Analysis and Partial Differential Equations, held September 22-23, 2012, at the Rochester Institute of Technology, Rochester, NY, USA. The book features new directions, results and ideas in commutative and noncommutative abstract harmonic analysis, operator theory and applications. The commutative part includes shift invariant spaces, abelian group action on Euclidean space and frame theory; the noncommutative part includes representation theory, continuous and discrete wavelets related to four dimensional Euclidean space, frames on symmetric spaces, $C DEGREES*$-algebras, projective multiresolutions, and free probability algebras. The scope of the book goes beyond traditional harmonic analysis, dealing with Fourier tools, transforms, Fourier bases, and associated function spaces. A number of papers take the step toward wavelet analysis, and even more general tools for analysis/synthesis problems, including papers on frames (over-complete bases) and their practical applications to engineering, cosmology and astrophysics.Other applications in this book include explicit families of wavelets and frames, as they are used in signal processing, multiplexing, and the study of Cosmic Microwave Background (CMB) radiation. For the purpose of organisation, the book is divided into three parts: noncommutative, commutative, and applications. The first group of papers are devoted to problems in noncommutative harmonic analysis, the second to topics in commutative harmonic analysis, and the third to such applications as wavelet and frame theory and to some real-world applications.
Author: Marc Aristide Rieffel
Publisher: American Mathematical Soc.
Published: 2004
Total Pages: 106
ISBN-13: 0821835181
DOWNLOAD EBOOKBy a quantum metric space we mean a $C DEGREES*$-algebra (or more generally an order-unit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov-Hausdorff di
Author: Volkher Bernhard Scholz
Publisher:
Published: 2012
Total Pages: 0
ISBN-13:
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Publisher:
Published: 2003
Total Pages: 768
ISBN-13:
DOWNLOAD EBOOKAuthor: Alain Connes
Publisher: Springer
Published: 2003-12-15
Total Pages: 364
ISBN-13: 3540397027
DOWNLOAD EBOOKNoncommutative 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.
Author: Ali Chamseddine
Publisher: Springer Nature
Published: 2020-01-13
Total Pages: 753
ISBN-13: 3030295974
DOWNLOAD EBOOKThis 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.