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$\DeclareMathOperator\coker{coker}$I think I'm in a pretty good position to answer this question because I am a graduate student working in noncommutative geometry who entered the subject a little bit …

I don't have any magical references for you, nor do I understand the NCG point of view on the Penrose tiling all that well. I learned just enough about this to convince myself that I didn't need to l …

First of all, let me mention that functional analysis plays a similar role in noncommutative geometry that commutative algebra plays in algebraic geometry, and it pays off to at least have a reference …

I would say that there are really only two definitions of K-homology commonly used in the literature (apart from the naive definition via the Bott spectrum): "analytic K-homology" and "geometric K-hom …

A very good place to start is Connes' book "Noncommutative Geometry", available for free on his website. It's a huge book, but it's possible to skip around quite a bit to get what you need. To begin …

KK-theory provides one way to eliminate the asymmetry in the definitions of K-theory and K-homology. A cycle in $KK(A, B)$ is a triple $(H, \rho, F)$ where $H$ is a (adjectives) Hilbert $B$-module, $ …

Probably the argument that you're looking for is based on the "Dirac / Dual Dirac" method. The idea is to exploit the product structure in KK-theory as much as possible - this makes the proof functor …

Let $M$ be a compact manifold and let $D$ be a minimal diffeomorphism from $M$ to itself (meaning there are no nontrivial invariant subspaces). I believe it was Connes who proved that if the first co …

Maybe it will help to see how the algebra $B$ is a special case of the general construction in $2.\alpha$. The compact manifold is the space $Y = \{a, b\}$ consisting of two points. The open cover o …