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I am a 3rd year undergraduate student. What are the things that would be good to know if I apply for PhD? I want to do a PhD in non-commuative topology.

I am fascinated by this non-commutative aspect. I have no knowledge of $K$-theory or $C^{\ast}$-algebras yet. I have read Hatcher upto homology. I am reading differential geometry fromm S. S. Chern and in analysis I am studying from Rudin's Real and Complex Analysis as well as Functional Analysis.

How much should I read in order to understand and from which books in order to understand non-commutative geometry fully?

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You may want to try reading something like Kalkhali's Very Basic Noncommutative Geometry arxiv.org/abs/math/0408416. Though in my (very limited) experience it's more effective to approach research by trying to tackle a specific problem and learning the tools you need as you go along. In lieu of that, read as much operator algebra and geometry as you can. –  mebassett Jul 5 '12 at 12:13
    
Depending on what flavor non-commutative topology you want to learn a lot of homological algebra. You might want to start glancing at Weibel's An Introduction to Homological Algebra. It is my understanding that a lot of motivation comes from derived categories of varieties. –  Matt Jul 5 '12 at 15:56
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Standard references for basic C*-algebra theory include:

C*-Algebras by Example by Kenneth R. Davidson

C*-Algebras and Operator Theory by Gerard J. Murphy

An Introduction to K-Theory for C*-Algebras by M. Rørdam, F. Larsen, N. Laustsen

In general, it seems safe to state that a solid knowledge of algebraic topology and functional analysis is useful for doing "non-commuative topology." If you have a background in algebraic topology, you might want to consider learning about K-theory in the commutative setting first, for example from Efton Park's Complex Topological K-Theory.

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