Wednesday, September 29, 2010

"Mass endomorphism, surgery and perturbations"

That is the title of an interesting paper just posted on the arXiv. I had never heard of the "mass endomorphism" so this was new to me... Take a compact Riemannian spin manifold and suppose that the metric is flat in the neighborhood of a point p.  If there are no harmonic spinors (so that the Dirac operator is invertible) then the Dirac Green's function, i.e. the inverse of the Dirac operator, has an asymptotic expansion near p in which the zero term is an endomorphism of the spinor bundle called the mass operator.  It is known that if the mass operator is non-zero then a solution exists to the classical Yamabe problem.  In this paper it is shown that the mass operator is "generically" non zero - using a lot of the machinery from positive-scalar-curvature land: psc surgery, results of Stolz, etc...

Thursday, September 16, 2010

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 for more details.

The deadline for applications is
-October 1, 2010 for Research Professors
-December 1, 2010 for Research Members and Postdoctoral Fellows.
Look at
for more details.

Permanence properties in coarse geometry

I hate to think how long it has been since I last posted here. My apologies - it has been a difficult summer for various non-mathematical reasons. Anyhow, trying to get back on track let me mention a survey article that Erik Guentner sent me called "Permanence properties in coarse geometry". What Erik means by "permanence properties" is statements like "the property of having finite asymptotic dimension is closed under group extensions". Many statements of this kind, for a variety of coarse properties (asymptotic dimension, embeddability in Hilbert space, property A/exactness, etc) have by now been proved and this is a very nice survey bringing together general techniques for obtaining such results with specific applications.