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I'm a Masters student currently deciding which area to focus on. So far, my primary interest has been $C^*$-algebras and operator algebras (I already have some knowledge of $K$-theory for $C^*$-algebras and Hilbert modules), but I always had some interest in geometry. When I learnt about the Gelfand Representation Theorem, I thought there was a chance to mix both interests.

I tried to read Alain Connes' Noncommutative Geometry and realized that I was missing lots of prerequisites. I'd like to have at least a broad enough picture of this area to determine whether I like it or not. What is the minimum I should know before I can do that?

user829347
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Julio Cáceres
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    Try reading this https://arxiv.org/abs/math/0408416 There is also a book by the same author which is much friendlier than Connes' book. I'd say given that you already know some Operator algebras and K-theory, probably some algebraic topology and differential geometry is useful. Try understanding the Atiyah Singer index theorem statement. – vap Nov 09 '17 at 19:23
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    I'm currently taking a course in diff geometry and took an introductory course for alg topology. Any book to study the Atiyah Singer index theorem? – Julio Cáceres Nov 10 '17 at 13:22
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    Heat Kernels and Dirac Operators Book by Ezra Getzler, Michèle Vergne, and Nicole Berline. The approach in this book is analytic. If you want a more K-theoretic approach there are two papers by Nigel Higson (One has K-theory in its name, the other deals with the tangemt groupoid) that you can google easily. – vap Nov 13 '17 at 02:41
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    Elliot Natsume Nest offer a different approach via the "classical limit". Trying to understand all of these proofs will keep you busy for a while and it'll get you to learn a lot around noncommutative geometry. – vap Nov 13 '17 at 02:50

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