Category Archives: My Publications

Contract signed for “Winding Around”

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.”

 

Paper on sheaves of C*-algebras and K-homology published

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.)

Positive scalar curvature partial vanishing theorems and coarse indices

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.

 

The real form of the coarse Baum-Connes conjecture

Following up on my post a few days back about Dranishnikov’s talk… After the talk, Sasha asked me if I knew a reference where some “standard” facts about the real version of the coarse Baum-Connes conjecture were stated (as, for example, that the real coarse index of the Dirac operator vanishes for positive scalar curvature manifolds, or that the complex form of the coarse Baum-Connes conjecture implies the real form.

I was sure that these “well known to experts” results must be written down somewhere. Maybe they are, but I couldn’t find a clean reference.  So I thought it might be helpful to put together a little note summarizing some of these standard facts.  I’ve now posted this on the arXiv and it is available here.  If you need the real version of CBC for something, this might be useful.

Originally, Nigel and I were going to cover the real version of everything in Analytic K-Homology.  But at some point we got fed up with Clifford algebras and retreated to the complex world.  I think that was the only way to get the book finished, but it has left a few loose ends!

Ghostbusting and Property A

Let \(X\) be a bounded geometry discrete metric space.  Guoliang Yu defined a ghost to be an element of the Roe algebra \( C^*_u(X) \) that is given by a matrix \(T_{xy}\) whose entries tend to zero as \(x,y\to\infty\).

The original counterexamples of Higson to the coarse Baum-Connes conjecture were noncompact ghost projections on box spaces derived from property T groups.  On the other hand, all ghost operators on a property A space are compact.

In Ghostbusting and Property A, Rufus Willett and I show that all ghosts on \(X\) are compact if and only if \(X\) has property A.  (Appropriately enough, on a space without property A we construct ghosts using the spectral theorem.) The paper will appear in the Journal of Functional Analysis.

Question: To what characterization of ordinary amenability does this correspond?