Noncommutative Geometry

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

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. ContentsPart I: BasicsModel examplesCategories and functorsC-algebrasPart II: Noncommutative invariantsTopologyAlgebraic geometryNumber theoryPart III: Brief survey of NCGFinite geometriesContinuous geometriesConnes geometriesIndex theoryJones polynomialsQuantum groupsNoncommutative algebraic geometryTrends in noncommutative geometry

Auteur
Igor V. Nikolaev, St. John's University, USA.

Résumé

"The book is a good survey of noncommutative geometry, and is an excellent starting point for doing good research in the field. Each section of the book ends with a list of references for going deeply into each subject, and so this book gives a framework for the field of noncommutative geometry."

Arvid Siqveland in: Zentralblatt für Mathematik 1388, 1-2

"In any case, it is true with a vengeance that NCG is a big and important subject, and the book we are looking at here, with its subtitle of "A functorial approach" is doubtless a very valuable contribution on any number of counts, not the least of which being that the more functorial we get, the more generality we reap, and the clearer the overarching structure becomes. [...] In point of fact, the book under review is the first book on NCG to take a functorial approach as its Leitmotiv, and that is in itself a very solid contribution, pedagogically and mathematically."

Michael Berg in: MAA Reviews (27.07.2018), https://www.maa.org/press/maa-reviews/noncommutative-geometry-a-functorial-approach