Author Archives: John Roe

Epsilon propagation and the Roe algebra

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.

 

 

K-theory course online

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.

 

AMS Open Math Notes

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!

 

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!