My review of Masoud Khalkhali’s *Basic noncommutative geometry* just appeared in the Mathematical Intelligencer. You can read it at the link below

http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s00283-014-9489-6

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My review of Masoud Khalkhali’s *Basic noncommutative geometry* just appeared in the Mathematical Intelligencer. You can read it at the link below

http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s00283-014-9489-6

So I signed the contract last week for “Winding Around”, my book based on the course I taught in the MASS geometry/topology track last year. It will appear in the American Mathematical Society’s *Student Mathematical Library* series, and the manuscript is due to be delivered to them on April 1st – I leave it to you whether or not you think this is an auspicious day! The book centers around the notion of “winding number” (hence “Winding Around”) and uses that as a peg on which to hang a variety of topics in geometry, topology and analysis — finishing up, in the final chapter, with the Bott periodicity theorem considered as one possible high-dimensional generalization of the winding number notion.

The intended audience is an undergraduate one (there was skepticism from some of the AMS readers about this, but I told them the MASS students made it through okay!) and the tone is, I hope, entertaining and discursive. As I say in the introduction, “*Winding around* is a description of the book’s methodology as well as of its subject-matter.”

I thought I’d finished this sequence of posts with number five, but then I spent a little time talking with Jerry Kaminker and Rufus Willett and I think that I understood two things: first, how to formulate the limit point construction more cleanly and, second, the “symmetry breaking” role of the ultrafilters which is not clear in what I had written so far. Read on. Continue reading

In the previous post I sketched out the condensation of singularities argument which finishes the proof under the assumption that the underlying metric space \(X\) is a group. In this case all limit operators act on the same Hilbert space, namely \(\ell^2(X)\), and the *weak compactness* of the set of all limit operators plays a critical role.

In the more general situation described by Spakula and Willett, each limit operator (say at a boundary point \(\omega\)) acts on its *own* Hilbert space \(\ell^2(X(\omega))\). In order to bring this situation under sufficient control to continue to make the weak compactness argument, we are going to need some kind of bundle theory. Continue reading

In this post I’ll finally get to the “condensation of singularities” argument that was invented by Lindner and Seidel in the (free abelian) group context and generalized by Spakula and Willett to metric spaces. (Calling this “condensation of singularities” is my idea, but it does seem to me to get at what is going on. I can’t help feeling that there should be a way of replacing some of the explicit constructions with an abstract argument involving the Baire category theorem. But I have not yet been able to come up with one.) Continue reading