Tag Archives: index theorem

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.


Michael Atiyah’s Birthday!

Heads up!  In  a couple of days (April 22nd) it is the 87th birthday of “Britain’s mathematical pope”, (not just Britain’s, either, IMO), otherwise known as my doctoral advisor, Professor Sir Michael Atiyah.   HAPPY BIRTHDAY MICHAEL!

To celebrate, his son David is assembling an online tribute – see http://www.atiyah.eu/mfa87/    Please consider sending a tribute message to david@atiyah.eu  Here’s what hes ays:

We are collecting messages of congratulations on the occasion of Michael Atiyah‘s 87th birthday Friday, April 22, 2016.

If you have the time, memory, and an inclination, please also include your favourite personal story about Britain’s Mathematical Pope*. I keep hearing every mathematician has one – it would be a shame not to collect and archive them for posterity.

Bonus points awarded for photographs, with prizes for the best MP4 video message we can share on the night.

Pls include:
– your name
– your current position, & location (if appropriate)
– when and where you first met Michael

We will keep it simple and hope to collate and publish submisssions in due course.

* = with thanks to Siobhan Roberts for the expression used in her recent biog of J H Conway – i have simply extended his Popedom from England to Britain.

If you haven’t seen it, here is a great article from Wired last week: Mathematical Matchmaker Atiyah Dreams of a Quantum Union.


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.


Macroscopic dimension and PSC, after Dranishnikov

Sasha Dranishnikov gave a talk describing some of his results about Gromov’s conjecture relating positive scalar curvature and macroscopic dimension.

Definition (Gromov) Let \(X\) be a metric space.  We say that \(X\) has macroscopic dimension \(\le n\) if there exists a continuous, uniformly cobounded \(f\colon X\to K\), where \(K\) is an \(n\)-dimensional simplicial complex.  We recall that uniformly cobounded means that there is an upper bound on the diameters of inverse images of simplices.

This is a metric notion, but it is quite different from the familiar asymptotic dimension.  One way of defining the latter says that \(X\) has asymptotic dimension \(\le n\) if, for each \(\epsilon>0\), there is an \(\epsilon\)-Lipschitz uniformly cobounded map to an \(n\)-dimensional simplicial complex (here, we agree to metrize \(K\) as a subset of the standard simplex in infinite-dimensional Euclidean space).  From this definition it is apparent that the macroscopic dimension is less than or equal to the asymptotic dimension.  On the other hand, it is also clear that the macroscopic dimension is less than or equal to the ordinary topological dimension.

Gromov famously conjectured that the universal cover of a compact \(n\)-manifold that admits a metric of positive scalar curvature should have macroscopic dimension \(\le n-2\).  The motivating example for this conjecture is a manifold  \(M^n = N^{n-2}\times S^2 \) – this clearly admits positive scalar curvature, and its universal cover has macroscopic dimension at most \(n-2\).  Gromov’s conjecture suggests that this geometric phenomenon is “responsible” for all positive scalar curvature metrics. Continue reading