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I recently came across noncommutative geometry and found it rather interesting. I should mention that I'm a graduate student considering options for my research and if I were to name an area which I'm interested in, then it would be functional analysis including operator algebras etc., and that was how I got to know about noncommutative geometry. From what I've been told, noncommutative geometry is a very broad area so I would think that one can go in many different directions after entering the field. What I would like to find out is what some of these directions are. Also, if I am to go into this field, my motivation will probably be functional analytic so I will be particularly interested to know if there is an approach that suits me.

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Please read: mathoverflow.net/howtoask To be more positive: don't you have in your department a professor / post-doc / more advanced PhD student with whom you could discuss this (rather vague) question over coffee? –  Alain Valette Apr 21 '13 at 16:40
    
i think so but I being a newbie myself I couldn't resist to pour my knowledge over –  Koushik Apr 21 '13 at 16:49
    
This should probably be community wiki, as you are asking for a list, rather than a single correct answer. You also accepted an answer pretty quickly! –  MTS Apr 21 '13 at 20:37
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2 Answers 2

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1)You look into the book "Noncommutative Dynamics and E0 semigroups" by William Arveson 2) There is an approach to attack multivariate operator theory through algebraic geometry.You may look to "Operator Theory and Complex Geometry" by douglas for an introduction 3)You may look into brown,douglas,fillimore's paper in essential normality and K-Homology which was a precursor to connes approach 4)or look into connes's book

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it's my personal opinion but I think the second approach may go a long way in the future.the heart of the approach is "arveson conjecture" which relates homogeneous varieties in the unit ball of $C^n $ with essential normality of d-shift operators vaguely –  Koushik Apr 21 '13 at 16:47
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Connes' book is pretty tough to get through as a beginner. I would suggest as an alternative the book Elements of Noncommutative Geometry by Gracia-Bondia, Varilly, and Figueroa. Or for a more concise, but less thorough, introduction, I like Varilly's book An Introduction to Noncommutative Geometry, which is also a little more recent.

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In view of your comment on OP: did you flag for moderators attention to make it CW. If not the procedure seems somewhat unfortunate since if OP turns the question into CW, your answer would stay non-CW (except moderators still notice it or you change it manually, of course). –  quid Apr 21 '13 at 20:59
    
@quid: Yes, he did. –  S. Carnahan Apr 22 '13 at 0:02
    
@S. Carnahan: Thanks for letting me know. –  quid Apr 22 '13 at 9:04
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