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I want to second David Ben-Zvi's recommendation of Nigel Higson's ICM lecture, though there is a bit of progress that has been published since it was written. Fortunately, you needn't venture further …

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 …

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 …

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 …

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 …

In what follows, all tensor products are graded. The comments about the existence of canonical (up to homotopy) $\ast$-homomorphisms $\mathbb{C} \to K(H)$ and $K(H) \otimes K(H) \to K(H)$ right befor …

[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\| …

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 …

$\DeclareMathOperator\coker{coker}$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 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 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 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 …

I hesitate to let this out, but there's always this cute little note that I learned from another MO answer (I don't know which one): https://www.maths.ed.ac.uk/~tl/glasgowpssl/banach.pdf. Maybe this …

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 …

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 …