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I am applying for a short undergraduate/honours research project, and am trying to decide on preferences for a topic.

One of the possible topics is noncommutative geometry. I like the sound of this, however I haven't really been able to find a self-contained introduction to the area (and perhaps for good reason).

My question is twofold:

  1. How much commutative geometry, if any, would you say is necessary before jumping into the noncommutative side of things? (I am an undergraduate just finishing my third year of study and so don't really have any prior exposure to algebraic geometry.)
  2. Would anyone be able to recommend some "introductory" expositions in this area?

Thanks heaps :)

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Geometry is a misnomer. Personally, I think that you would struggle. It involves a lot of general mathematics, especially category theory. In the usual setting, you can consider the space of functions on a space. These will often form an algebra. Moreover, given the algebra, it is sometimes possible to recover the space. The algebras are usually commutative. What if you have a noncommutative algebra? What space gave rise to this as a space? If you haven't studied ordinary algebraic geometry then I think you might die a slow and painful death trying to work with noncommutative geometry. –  Fly by Night Aug 28 '12 at 22:45
    
There are several types of noncommutative geometry. I think the one is slightly more accessible than the the other. The one I'm familiar with is completely based off generalizing the bounded derived category of coherent sheaves on a scheme. It would be useful to be familiar with that in order to understand what in the world the motivation is...so I guess I'm saying I agree with Fly by Night. –  Matt Aug 28 '12 at 23:06
    
Have you had abstract algebra or differential geometry in your course work? How serious is this project? It's hard to say where to start if we don't know where you are or how carefully you must walk the road ahead. –  James S. Cook Aug 29 '12 at 0:10
    
How about this: arxiv.org/pdf/gr-qc/9906059v2.pdf –  James S. Cook Aug 29 '12 at 0:30
    
I noticed he also has a book. Might be worth looking at "An Introduction to Noncommutative Differential Geometry and its Physical Applications", however, this is more in the physicsy direction. Maybe you want a pure math view on this topic? –  James S. Cook Aug 29 '12 at 0:35

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