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

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.”

After a busy day giving final exams in the MASS program it was nice to learn today that my paper with Paul Siegel about sheaves of C*-algebras has appeared in the *Journal of K-Theory*. The link for the published version is

http://journals.cambridge.org/repo_A91rlWyM

This paper arose from some discussions when Paul was writing his thesis. We were talking about the “lifting and controlling” arguments for Paschke duals that are used in the construction of various forms of operator-algebraic assembly maps (an early example is the one that appears in my paper with Nigel on the coarse Baum-Connes conjecture, which asserts that the quotient \( D^*(X)/C^*(X) \) of the “controlled” pseudolocal by the “controlled” locally compact operators does not depend on the assumed “control”). At some point in these discussions I casually remarked that, “of course”, what is really going on is that the Paschke dual is a sheaf. Some time later I realized that what I had said was, in fact, true. There aren’t any new results here but I hope that there is some conceptual clarification. (There is an interesting spectral sequence that I’ll try to write about another time, though.)

This paper, http://arxiv.org/abs/1210.6100, has been accepted by the Proceedings of the Edinburgh Mathematical Society. I just sent off the copyright transfer form this evening, so everything is now set, I hope.

The paper is mostly paying an expository debt. In my CBMS lecture notes I said that if one has the Dirac operator on a complete spin manifold \(M\), and if there is some subset \(N\subseteq M\) such that \(D\) has uniformly positive scalar curvature outside \(N\), then the index of \(D\) belongs to the K-theory of the ideal \(I_N \triangleleft C^*(M) \) associated to the subset \(N\). A very special case of this is the observation of Gromov-Lawson that \(D\) is Fredholm if we have uniformly positive scalar curvature outside a compact set. There are of course analogous results using thepositivity of the Weitzenbock curvature term for other generalized Dirac operators.

Until now, I had not written up the proof of this assertion, but I felt last year that it was (past) time to do so. This paper contains the proof and also that of the associated general form of the Gromov-Lawson relative index theorem which also appears in my CBMS notes. The latter proof uses some results from my paper with Paul Siegel on sheaf theory and Pashcke duality.

The submission to PEMS is in honor of a very pleasant sabbatical spent in Edinburgh in fall 2004.