We hope to submit the final manuscript for the book *Mathematics for Sustainability* to Springer in a couple of weeks. There is a web site linked to the book: https://math-for-sustainability.com

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We hope to submit the final manuscript for the book *Mathematics for Sustainability* to Springer in a couple of weeks. There is a web site linked to the book: https://math-for-sustainability.com

I remember with great pleasure giving the CBMS lectures that became the book *Index Theory, Coarse Geometry and the Topology of Manifolds . * I was at the beginning of a six month visiting appointment, delighting in the change to be with my family in this beautiful city, and energized by coming up with two lectures every day (and I mostly wrote the book as we went along – how nice to be young and energetic).

However this speedy process meant that one (or maybe more) over-optimistic statements slipped by. One in particular that has caused trouble over the years is the *Remark* after Lemma 3.5 In this Remark I define a notion of \(\epsilon\)-propagation (notation being as in the usual setup for Roe algebras):

**Definition:** The \(\epsilon\)-propagation of an operator \(T\) is the infimal \(R\) such that

\[ \|f\|\le 1, \ \|g\|\le 1, \ d(\text{Support}(f),\text{Support}(g))\le R \Longrightarrow \|fTg\| \le \epsilon. \]

It is clear that an operator in the Roe algebra has finite \(\epsilon\)-propagation for all \(\epsilon\). It is also clear (though I didn’t point this out at the time) that the collection of operators having finite \(\epsilon\)-propagation for all \(\epsilon\), which some people call *quasi-local* operators, is a \(C^*\)-algebra. So the question arises: is it the same as the Roe algebra?

So here’s where my optimism came in. I causally remarked, “If \(X\) is large scale finite dimensional—by which I meant what we would now call finite asymptotic dimension—then the converse is the case” (“the converse” being the statement that every quasi-local operator is in the Roe algebra). I *thought* that there would be a fairly obvious “large scale patrition of unity” proof—and somehow never checked. There isn’t. *Mea culpa*. Over the years people proved the statement for \(\mathbb Z\) and then \({\mathbb Z}^n\), but these proofs used Fourier series and clearly didn’t generalize.

Until this year. Back last summer I received a note from Aaron Tikuisis of Aberdeen asking about the status of the statement and saying that he had a proof. This has now led to a preprint by Tikuisis and Spakula showing the equivalence not just for spaces of finite asymptotic dimension but also of “finite decomposition complexity” (see Guentner, Tessera and Yu, *A notion of geometric complexity and its application to topological rigidity*, **Inventiones **189 (1012), 315-37). They say it will be on the arXiv soon. I just had another inquiry from a grad student about this misleading remark of mine and was happy, this time, to be able to point him to an honest proof.

If only I’d called it a “conjecture”! Still, I am very happy that the question is resolved in such a nice way. The paper uses some of the new ideas in classification, like nuclear dimension, though I have not read it in enough detail to be able to explain how.

My C*-algebra notes (from the Fall 2015 course) are now on *AMS Open Math Notes*. You can find them here.

I am putting the notes from my K-theory course online as they are available. You can find them here.

I am starting with a purely algebraic development, as in the first few chapters of Milnor, but will soon dramatically change gear and talk about C*-algebras.

We had the interesting surprise of having a TV reporter in our class the other day. You can view his report/interview here.

I wonder if you know about the AMS *Open Math Notes* project? I only just heard of it, but I feel very positive about any projects that make mathematical content more freely available. Here is the link to the AMS page about the project:

http://www.ams.org/open-math-notes/omn-about

As my students know, I’ve made a habit over the years of putting together TeXed notes for the courses I deliver – especially graduate courses – and now I have quite a number of them. With some pressure on my time (read: cancer) there is no way that I could bring all of these to formal publication, even if that was the right route for them. But as “MathNotes” I can see that they might be helpful. So I’m going to start submitting them, perhaps after light revision, to the AMS site. I made a start today by posting my notes from the Penn State complex analysis (graduate) course, which I’ve delivered three or four times, taking a slightly different tack each time. Based on what I learn from that, I have a good queue of other notes to submit as well. This is actually quite exciting for me.

*Updated, December 20th*: The notes have now appeared on the OpenMathNotes site, and may be found here.

In further exciting (to me) AMS news, my *Winding Around* made it to their 2016 bestseller list! Because of this, the AMS is offering a special discount for orders placed between now and the end of January…

I hope to upload further packages of notes in the new year! Best wishes to all!