Open Questions: Noncommutative Geometry

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See also: Geometry and topology -- Mathematics and physics -- M-theory -- Quantum gravity -- Quantum geometry

Introduction

Noncommutative geometry and the Riemann zeta function

Hopf algebras and quantum groups

M(atrix) theory


Recommended references: Web sites

Recommended references: Magazine/journal articles

Recommended references: Books

Introduction



Recommended references: Web sites

Noncommutative geometry
Article from Wikipedia.
Noncommutative Geometry
"This site has been designed in order to offer to the community of researchers in Noncommutative Geometry a center of resources related to their work. Please read the Help file if you want to provide information. The main purposes are to display Job offers, conference announcements, and papers which can be of interest for noncommutative geometers, from the prospective to specialized levels. The information presented on this site is provided by researchers themselves."
The Geometer of Particle Physics
July 2006 Scientific American article. "Alain Connes's noncommutative geometry offers an alternative to string theory. In fact, being directly testable, it may be better than string theory."
What are C*-algebras good for?
Brief overview giving physical motivation without mathematical details, by John Baez.
Noncommutative Geometry and Quantum-Group Theory
Very brief summary of the field, from a tutorial conference on the subject.
Geometry in Physics Tomorrow
Slide presentation given by Isadore Singer at the 2001: A Spacetime Odyssey conference.
Nonabelian Gauge Theories on Noncommutative Spaces
Slide presentation given by Bruno Zumino at the 2001: A Spacetime Odyssey conference. Includes remarks on the Seiberg-Witten equations.


Recommended references: Magazine/journal articles

Quantum Spaces and Their Noncommutative Topology
Joachim Cuntz
Notices of the AMS, September 2001, pp. 793-799
Noncommutative geometry is based on the idea that the algebra of functions on a topological space contains information about the space. When the algebra is noncommutative, the result is noncommutative geometry. The goal is to derive "topological" properties of a space from its corresponding algebra of functions. The two fundamental techniques available to do this are cyclic homology and topological K-theory.
[Article in PDF format]
Noncommutative Curves and Noncommutative Surfaces
J. T. Stafford; M. Van den Bergh
Bulletin of the AMS, April 2001, pp. 171-216
Classical algebraic geometry is thoroughly entwined with the theory of commutative rings. Nevertheless, a large part of the subject can be generalized to the category of noncommutative rings, though not in a straightforward manner.
[Abstract, references, downloadable text]
Review of Noncommutative Geometry by Alain Connes
Vaughan Jones; Henri Moscovici
Notices of the AMS, August 1997, pp. 792-799
Although presented as a book review, this is really an expository paper on noncommutative geometry as developed by Connes. The basic idea is that in some situations which have been treated geometrically, the geometry itself can be better understood as a study of a noncommutative algebra than as properties of a set of points.
[Article in PDF format]
Noncommutative Geometry
Andrew Lesniewski
Notices of the AMS, August 1997, pp. 800-805
Quantum field theory is explicitly a geometrical approach to the physical problems it deals with. This article explains how Connes' noncommutative geometry can clarify a lot of modern field theory.
[Article in PDF format]
Review of Topics in non-commutative geometry by Yuri I. Manin
Ivan Penkov
Bulletin of the AMS, July 1993, pp. 106-111
The field of noncommutative geometry has recently been dominated by Connes' work emphasizing C* algebras, but its history goes back well before that. Manin's book deals with many topics, but especially "supergeometry" and quantum geometry (Hopf algebras).


Recommended references: Books


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