## Noncommutative geometry?

How new is this subject of noncommutative geometry? I tried googling it, but few info comes out and there is not alot of books about it either.

What is this subject about exactly and is it going to be something major?
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 Quote by waht How new is this subject of noncommutative geometry? I tried googling it, but few info comes out and there is not alot of books about it either. What is this subject about exactly and is it going to be something major?
Hi, I think the subject is around 30 years old (it might be even more). Intuitively, non commutative geometry is a strict algebraic theory that allows one to generalize Riemannian manifolds. Connes remarked that such a structure (actually, you have to restrict yourself to manifolds with a spin structure if I remember correctly) can be fully characterized by the *commutative* C* algebra of C^infty functions equipped with a derivative operator. Now, you can ask yourself what geometry´´ you get when you allow the C* algebra to be non commutative. At that point you can use the GNS representation theorems wich say that such non abelian C* algebra can be represented in terms of bounded operators on some Hilbert space. This gives you a link with quantum mechanics and one could hope to get quantum gravity out in this way. If you want references: search on Connes first.

Cheers,

Careful
 Recognitions: Homework Help Science Advisor Non-commutative geometry is a blanket term: Connes definition, if he even has such a thing as 'a definition' would not agree with, say, an algebraic geometer's idea. The first thing you should ask yourself is: do i know what commutative geometry is? If so then it is relatively easy to see what 'non-commutative' geometry is: geometry without the restriction of commutativity. How you relax that criterion would I suspect depend upon whom you asked.

## Noncommutative geometry?

 Quote by matt grime Non-commutative geometry is a blanket term: Connes definition, if he even has such a thing as 'a definition' would not agree with, say, an algebraic geometer's idea. The first thing you should ask yourself is: do i know what commutative geometry is? If so then it is relatively easy to see what 'non-commutative' geometry is: geometry without the restriction of commutativity. How you relax that criterion would I suspect depend upon whom you asked.
Indeed, and I gave one which is used by physicists (and which I remember to have read from a paper Connes has written for physicists). More abstract stuff can be found on webpages of Lieven Lebruyn and Michel Van den Bergh.

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