Noncommutative Geometry Explained

[waht]

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

What is this subject about exactly and is it going to be something major?

 

[Careful]

How new is this subject of noncommutative geometry? I tried googling it, but few info comes out and there is not a lot 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 which 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

 

[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.

 

[Careful]

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.