# “Finite part of operator K-theory” IV

Continuing this series (earlier posts here, here and here) on the paper of Weinberger and Yu, I’m expecting to make two more posts: this one, which will say something about the class of groups for which they can prove their Finite Part Conjecture, and one more, which will say something about what can be done with the conjecture once one knows it. Continue reading

# Thinking about “maximal Roe algebras”

One of the things that has happened in coarse geometry while I was busy being department chair is a bunch of papers about “maximal Roe algebras” (some references at the end). Of course these are objects that I feel I ought to understand, so I spent some time trying to figure out the basics.

Let $$X$$ be a bounded geometry uniformly discrete metric space.  (Something like bounded geometry seems to be necessary, for a reason that I’ll explain below.)  We know how to form the translation algebra $${\mathbb C}[X]$$ (the *-algebra of finite-propagation matrices on $$X$$ ), and this has an obvious representation (the regular representation) on $$\ell^2(X)$$.  Then the usual version of the (uniform) Roe algebra is just the C*-algebra obtained by completing $${\mathbb C}[X]$$ in this representation.  Because it involves only the regular representation we may call this the reduced Roe algebra (in analogy to the group case). Continue reading

# 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!

# Imported “Interesting Mathematics”

For a few years I ran a blog over at blogspot.com called “Interesting Mathematics”, the URL being coarsemath.blogspot.com. It went quiet in 2010, which was a high-stress year for me, and never really got started again.

So, I succeeded (just now) in figuring out how to import all the old posts from “Interesting Mathematics” onto this blog. Of course that gives me an incentive to continue blogging about mathematics here.

# Coarse Math at MSRI!

Forwarded from Vincent Lafforgue

From August 15, 2011 to December 16, 2011, MSRI (Berkeley) hosts a program on Quantitative Geometry. It is organized by Keith Ball, Emmanuel Breuillard, Jeff Cheeger, Marianna Csornyei, Mikhail Gromov, Bruce Kleiner, Vincent Lafforgue, Manor Mendel, Assaf Naor (main organizer), Yuval Peres, and Terence Tao. This is a big program with many available positions.

Examples of areas that will be covered by the program are: geometric group theory, the theory of Lipschitz functions (e.g., Lipschitz extension problems and structural aspects such as quantitative differentiation), large scale and coarse geometry, embeddings of metric spaces and their applications to algorithm design, geometric aspects of harmonic analysis and probability, quantitative aspects of linear and non-linear Banach space theory, quantitative aspects of geometric measure theory and isoperimetry, and metric invariants arising from embedding theory and Riemannian geometry. Go to http://tinyurl.com/28x94y6 for more details.