Talk:Noncommutative geometry

Untitled

I don't entirely understand what the technical edit template is for in this case. The subject is very complicated, so the article is going to be technical if it is accurate. The adjectives are needed or the mentioned theorems may not work - I don't think we should try to get rid of them or describe them with "verbs", as every such adjective can be linked to an article explaining what it means. What I would push for is better exposition and more examples to motivate the topic. - Gauge 05:42, 3 September 2005 (UTC)

I agree with you, and feel that the article has come far enough that the template should be removed. One thing though, you should take a look at the article before the template was added. It was full of conversational sentences that were not appropriate, like: "What about the case of differential manifolds? Well...". The editing that you've done in your most recent edit, in my humble opinion, pushes this article over the edge so that it no longer has that 'needs editing' feel. - Tristanreid 20:41, 3 September 2005 (UTC)

Thank you. I will go ahead and remove the template then. - Gauge 23:58, 3 September 2005 (UTC)

Further Reading

I know Wikipedia isn't supposed to be a grocery list of references or whatever, but I feel that these resources are useful for people with the mathematical background of abstract algebra and differential geometry (speaking from experience...) so I wanted to share the papers:

• Noncommutative Geometry Year 2000 by Alain Connes

• Trace formula in noncommutative geometry and the zeros of the Riemann zeta function by Alain Connes

• A Short Survey of Noncommutative Geometry by Alain Connes

• Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples by Alain Connes and Michel Dubois-Violette

• Non commutative finite dimensional manifolds II. Moduli space and structure of non commutative 3-spheres by Alain Connes and Michel Dubois-Violette

• SU(n)-Connections and Noncommutative Differential Geometry by Michel Dubois-Violette and Thierry Masson

• On Curvature in Noncommutative Geometry by M. Dubois-Violette, J. Madore, T. Masson, and J. Mourad

• Some aspects of noncommutative differential geometry by Michel Dubois-Violette

I found these useful for studying the subject, but others may not. Whatever. Pqnelson (talk) 00:53, 27 December 2007 (UTC)

Also, the quantum calculus dictionary which is defined by Connes has use of the Dixmier trace, which has few resources dedicated to it. I found this and a student studying this would find this to be an invaluable resource:

• Dixmier traces and some applications to noncommutative geometry by A.L. Carey and F. A. Sukochev

There is no wikipedia page on the Dixmier trace and I'm not sure if it deserves one, so I thought this may be a good alternative. Pqnelson (talk) 17:21, 28 December 2007 (UTC)

NC manifolds

I entirely removed the section on NC manifolds as it was:

Another area of study is that of non-commutative differentiable manifolds. An ordinary differentiable manifold can be characterized by the commutative algebra of smooth functions defined on the manifold, and the space of smooth sections of its tangent bundle, cotangent bundle and other fiber bundles. All these spaces are modules over the commutative algebra of smooth functions. The concepts of exterior derivative, Lie derivative and covariant derivative are also important elements in understanding derivations over this algebra. In the non-commutative case, the algebras in question are non-commutative. To handle differential forms, one must work with the graded exterior algebra bundle of all p-forms under the wedge product and look at its algebra of smooth sections. A "differential" is taken to be an antiderivation (or something more general) on this algebra which increases the grading by 1 and is quadratically nilpotent.

The reasons for doing so are:

1.There was no mention of spectral triples 2. Some of the statements seem to be wrong (e.g. charactarization of smooth manifolds by the algebra of functions and some vector bundle...), although I might be worng in stating this. 3.The remarks on forms and exterior derivatives are very unclear, and not mention Hochschild and cyclic cohomology, which as far as my knowledge reaches are the appropriate generalizations of forms and De Rham cohomology in the operator algebraic setting.

Alterationx10 (talk) 20:03, 12 July 2008 (UTC)

Giovanni Amelino-Camelia

Is this Italian physicist an important figure in the field, as his biography suggests? Or is he non-notable, or a fringe theorist, or both? Thanks for any advice. Itsmejudith (talk) 10:54, 22 September 2009 (UTC)

He is very important for reflection about the role of some specific space-times in deformations of special and of general relativity and study of such specific models, for example of kappa-Minkowski space time. He did not contribute much to the general mathematical theory of noncommutative geometries, but rather to the task of singling out some physically sound models and finding some interesting detail about them. ' Zoran.skoda (talk) 15:26, 13 September 2010 (UTC)

Copyright problem

This article has been revised as part of a large-scale clean-up project of multiple article copyright infringement. (See the investigation subpage) Earlier text must not be restored, unless it can be verified to be free of infringement. For legal reasons, Wikipedia cannot accept copyrighted text or images borrowed from other web sites or printed material; such additions must be deleted. Contributors may use sources as a source of information, but not as a source of sentences or phrases. Accordingly, the material may be rewritten, but only if it does not infringe on the copyright of the original or plagiarize from that source. Please see our guideline on non-free text for how to properly implement limited quotations of copyrighted text. Wikipedia takes copyright violations very seriously. --r.e.b. (talk) 17:12, 24 December 2009 (UTC)

History

Section History has been in the version which I found only about some historical motivation from ergodic theory so I have removed the section History until somebody writes such an appropriate section and created instead a subsection on motivations from ergodic theory under section Motivation. In addition to much material on various approaches to schemes and general introduction I have seriously updated the references and moved few sentences between different sections in order to have more logical organization of the article. --Zoran.skoda (talk) 15:26, 13 September 2010 (UTC)

Tetrads

I removed the remark about the noncommutativity of some vector fields in general relativity as the remark does not propose any noncommutative space. It is about a fact on vector fields on a commutative space. This can be used possibly to construct some new noncommutative constructions but I do not see such a proposal written, so it is off topic. --Zoran.skoda(talk) —Preceding undated comment added 15:19, 25 October 2011 (UTC).

Motivation

The first part of this article surprises me quite a bit. It says about the motivation of the subject that it is to find a duality between noncommutative algebras and "geometric structures of certain kind". This is really not the case as the starting point of the subject is that such a duality is NOT there. The goal is to generalize geometric methods to the more general setting of noncommutative algebras in order to study THOSE. The geometric methods will have no geometric meaning, as there is no space of any kind present, but they will help us in understanding the algebra in question. I will go ahead and rewrite the introduction in the near future. Alterationx10 (talk) 23:10, 28 November 2011 (UTC)

Leading direction

'I removed a sentence from the introduction suggesting that the field was led by one particular mathematician. Is there support from independent reliable sources for such an assertion? We don't usually make such value judgements so it would have to be particularly well-supported. Deltahedron (talk) 10:43, 10 August 2014 (UTC)

I agree with the removal, but I would like to say that the opinion is rather common. The AMS classification's mention of Connes's name in the subject of noncommutative geometry might be worth noting. Minimalrho (talk) 21:41, 11 August 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Noncommutative geometry/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

At the moment, reads very much like a disambiguation page. Basically, one idea is regurgigated (lack of commutativity), but the article does not convey an accurate impression about either the goals or the methods of noncommutative geometry. Arcfrk 04:37, 15 June 2007 (UTC)

Last edited at 04:37, 15 June 2007 (UTC). Substituted at 02:24, 5 May 2016 (UTC)