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Here is a very rough outline of the proof of the index theorem using KK-theory: Define $KK_G(A, B)$, where $G$ is a Lie group and $A$ and $B$ are [adjectives] C*-algebras, and the Kasparov product be …

[Updated to include Nik Weaver's correction / improvement from the comments.] I think this follows from the spectral radius formula: $$\left\| a \right\|^2 = \rho(a^*a) = \lim_{k \to \infty} \left\| …

My personal favorite example of this phenomenon occurred in 1931. On one hand, Dirac published the paper Quantized Singularities in the Electromagnetic Field which hypothesized the existence of what …

I'll start with a meta-answer: given the large (and growing) number of programming languages out there, how do you decide where and how to invest your time? The answer turns out to be quite simple, b …

If you're willing to compromise on the operator algebras part then "Introduction to Hilbert Spaces with Applications" is close to optimal. It includes not only a detailed discussion of the spectral t …

I would check out "Heat Kernels and Dirac Operators" by Berline, Getzler, and Verne. It covers quite a bit of territory: -Characteristic classes: Much stronger than most books; develops Chern-Weil t …

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 …

I don't have my copy handy, but I think this is all worked out in chapter 2 of Naber's "Topology, Geometry, and Gauge Fields: Interactions". The book is essentially a textbook on differential geometr …

I think the books "Topology, Geometry, and Gauge Fields: Foundations" and "Topology, Geometry, and Gauge Fields: Interactions" are exactly what you're looking for. They are not short, but you don't a …

I think the main topological significance of the element you identify is its close relationship with the Thom isomorphism $K(M) \cong K(T^*M)$. I would imagine that there is also some connection with …

One common way to quantify non-commutativity which is especially popular in operator algebra theory and non-commutative geometry is to use the Schatten norms. Given a bounded operator $T$ on a separa …

I'm looking for a book, article, or lecture notes that does basic cohomology theory from a relative point of view (including the Thom isomorphism, the excision theorem, Lefschetz duality, the Gysin se …

$\DeclareMathOperator\coker{coker}$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 really strongly recommend chapter 2 of Naber's "Topology, Geometry, and Gauge Fields: Interactions". In this book and its companion volume "Topology, Geometry, and Gauge Fields: Foundations", Naber …

After writing this, I noticed that Donu Arapura made essentially the same point in a comment. But perhaps my added detail will be of use. There are two missing ingredients in what you wrote. Firs …