# Coarse Index Theory Lecture 2

Here is the follow-up lecture (second of two) on coarse index theory. I tried to bear in mind that the conferees in Germany had heard quite a few presumably much more detailed presentations in between by lectures 1 and 2, so I attempted to give a fairly “big picture” overview.  I had prepared to talk about several examples that I didn’t have time to discuss, so you will find some slides at the end of the presentation below that were not talked about in the video.

Here’s the video of Lecture 2:

And here is the link to the corresponding slides. Hope you find the presentation helpful and enjoyable!

# Coarse Index Theory Lecture 1

I gave the first of the two coarse index theory lectures yesterday.  The Polycom equipment makes a recording as standard, and I have uploaded it to YouTube.   So, you can take a look.  Is this an effective way to communicate mathematics?  It seemed to me to work pretty well.

I reviewed the basic definitions of the coarse index and then presented the always-elegant example of the partitioned manifold index theorem.  It seemed as though the presentation could be followed well enough by the German audience; only the business of asking and answering questions was a bit clunky.  Here is a direct link to the slides.

# 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.

# Maximal Roe algebras, part 3

It is a well-known fact that if a group $$\Gamma$$ is amenable then the canonical map $$C^*_{\max}(\Gamma) \to C^*_r(\Gamma)$$ is an isomorphism.  In this post I’ll follow Spakula and Willett in proving the coarse analog of this statement: for a property A space the canonical map from the maximal to the reduced Roe algebra is an isomorphism.  (The converse is unknown, unlike in the group case.) Continue reading