My talk at BIRS

I gave a talk yesterday (August 8th, 2013) on Ghostbusting and property A.  Thanks to the technology system at BIRS you can watch the talk on video here.

The paper has now been accepted for the Journal of Functional Analysis.

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

Cutting a sphere in half

I’m at the Banff International Research Station this week for a conference on metric geometry.   I’ve listened to several nice talks already but one that stood out for me was by Yevgeny Liokumovich on the problem of cutting a sphere in half.  (It had, of course, a more official title!)

Consider the sphere $$S^2$$ with some Riemannian metric, scaled so that the total area is 1.  Is there an upper bound to the length of a geodesic loop that divides the sphere into two disks of equal area?

It seems plausible at first that the answer might be “yes”, but in fact it is “no”.  To see the counterexample, think about balloon animals: specifically a “balloon starfish” that has three thin, cylindrical arms of length $$\ell$$ emanating from a central core. Continue reading

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.