Talking More, Flying Less: Coarse Index Theory Lectures at Freiburg

At the beginning of this year (which now seems a very long time ago) I accepted an invitation from Thomas Schick to speak in a week-long summer school at Freiburg on the subject of “Coarse methods in index theory”.   This was before the upheavals began in my life this year, one of which has been a severe illness making it impossible for me to travel.

From one point of view this is big disappointment, but from another it’s an opportunity.  I’ve long been worried by the inconsistency between the “green” values embraced by many academics, including me, and the ease with which we seem to justify jetting round the world to talk to one another about “Coarse methods in index theory”, or whatever it may be.  I totally agree that there are things we can learn face-to-face which are much harder to learn through alternative media.  Some conferences have been life and career changing for me.  But it’s hard to argue that this justifies every conference, in the face of the existential threat posed by climate change. Such is the argument made by the group of academics at flyingless.org:

Flying is an elite activity. The vast majority of the world’s population has never flown. Academics–particularly those from the world’s most prosperous countries–fly more frequently than most people do. University communities typically embrace sustainable practices in other areas of daily life. It would be inconsistent to ignore sustainability just in the case of flying.

University-based faculty, staff, and students can make large reductions in their total greenhouse gas emissions with moderate sacrifice in terms of institutional goals, professional advancement, and quality of life. However, they require mechanisms that are institutionally sensitive to differences in status, power, and position, as well as the right structural supports.

So I have an opportunity to try to implement a “flyingless” policy by delivering my lectures using remote streaming technology.  There will be two lectures which I’ll give to an audience at Penn State and which will also be livestreamed to Freiburg (assuming we can all figure out the technology in time).  The Freiburg audience will have the opportunity to interact with the speaker (me) as well as with the Penn State audience.    By not sending me physically to Germany, we save $$2\frac12$$ tons of carbon dioxide emissions (a conservative estimate) and this compares with the 4 tons per person per year which is the current global average. There may be some disadvantage to the conferees in not having me physically present but I would guess it’s small. We shall see!

Here’s a schedule of the talks for those who are interested.  It is possible that we may be able to make the stream public – in which case I’ll post the information here so anyone can watch!

Lecture 1: Title: Coarse geometry and index theory

Abstract: I will try to explain why there is a close connection between the underlying idea of coarse geometry (that geometric information is encoded in the “large scale structure” of metric spaces) and the underlying idea of index theory (that topological information is encoded in the “low energy structure” of elliptic operators).  This lecture will be livestreamed to the Coarse Index Theory conference in Freiburg, Germany (and the audience there will participate by livestream too).

Lecture 2:  Title: Coarse geometry and structure invariants

Abstract:  If an elliptic operator has index zero, then it is “stably invertible”.   The reasons for such stable invertibility can themselves be analyzed and classified; they are called analytic structures associated to the operator in question.  In this talk I’ll give an introduction to the theory and application of analytic structures.   This lecture will be livestreamed to the Coarse Index Theory conference in Freiburg, Germany (and the audience there will participate by livestream too).

Marc Rieffel 75th birthday proceedings

I just received a message about the publication of a set of conference proceedings honoring Marc Rieffel’s 75th birthday.  The proceedings originate from a program at the Fields Institute that was held last year (June, 2013).

To quote the web site of the proceedings: This special issue is a tribute to Marc A. Rieffel, marking (approximately) his 75th birthday. It is the outgrowth of a Focus Program on Noncommutative Geometry and Quantum Groups that took place at the Fields Institute for Research in Mathematical Sciences in Toronto, Canada, in June, 2013. Marc Rieffel has been one of the most influential researchers in the world in the areas of noncommutative geometry and quantum groups. He has had over 30 PhD students and over 80 “mathematical descendants”. Among his major contributions were the introduction of Morita equivalence as a fundamental notion in noncommutative geometry and the classification of C*-algebras, the introduction of stable rank as a basic invariant of C*-algebras, the introduction of strict deformation quantization to construct new examples of quantum groups, and the analysis of the metric structure of noncommutative geometries. The papers in this special issue reflect the wide range of his contributions to mathematics as well as the great esteem in which he is held by the world mathematical community.

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