What are some ideas on the fastest way to get from undergrad background in math to remotely reasonable understanding of noncommutative geometry.

Question

What would be the fastest way to get from half of a class worth’s of (graduate) measure theory, same of functional analysis and generalized functions (or distributions), and only differential geometry from general relativity and quantum field theory classes to a barely decent understanding of noncommutative geometry. I also have a bachelors in math. (And, I took an easy graduate class in complex analysis – part 1 of two part course). But, maybe my bits and pieces of graduate math knowledge should be ignored to better serve people also looking for what I have in the title (meaning sticking strictly to just undergrad background)?

It seems the normal trajectory is something like: real and complex analysis, functional analysis, topology, differential geometry, algebraic geometry, and then noncommutative geometry? What are the smallest pieces that I can take from these (or whatever the trajectory is) to get there with the background would I need but minimizing unnecessary stuff along the way?

Also, if this isn't the right place to ask this, or if my tags are not good (or anything else), can you please let me know what to do instead?

Edit: The main question here is "What are the smallest pieces that I can take from these (or whatever the trajectory is) to get [to my first book on NCG]? If you can answer this (or if it's in your links), I will accept your answer (and many thanks to you). If you can also say which books (and which parts to skip), and which NCG book to go with after all of said books, that would be great, and I imagine you'll get a lot of upvotes from undergrads (and probably a lot of the curious people searching around too). (Or, textbooks that happen to avoid parts from the books on real, complex, and functional analysis and differential geometry (and w/e else) that weren't intended for a direct path to NCG.)

Thanks,

Answer 1

Unnecessary is a tricky concept. If you plan to do research you never know what pieces of knowledge will become useful.

That being said, if you only want to understand theorems in noncommutative geometry, take real and complex analysis and functional analysis, then get yourself a book 'introduction to $C^*$-algebras' or something like that.

Differential geometry is essentially the commutative geometry where noncommutative geometry comes from. So this is very useful to get an intuition to what people are trying to do and prove with noncommutative geometry. Algebraic geometry to me is also the idea to generalize geometry to a different setting but as far as I know it doesn't have many direct connections to noncommutative geometry.

Answer 2

The fastest way may be dangerous as you risk to go off course, or to miss the turn (on the trajectory).

Let me propose a question as an answer:
Why not reversing the arrow by choosing a target question in NCG, and deduce from it what is relevant for its formulation and understanding?
That approach also appears to reduce the risk to get stuffed with "unnecessary stuff along the way" (quoting from your post).

Concrete instances could be the Baum-Connes conjecture or the Kadison-Kaplansky conjecture (both dealing with $K$-theory for group $C^∗$-algebras), which are open. In particular, this proves that NCG is not a settled field. For further suggestions see e.g. the older Theories of NCG on MathOverflow.

In general there's the survey by Joachim Cuntz, 18 years ago in the Notices of the AMS and still a good read!
And here on math.SE, I'd consider Michael's answer to be helpful.

Further links which may be of interest

• older question on math.SE
• nLab on NC algebraic G