Category Archives: Research

The Witness Relocation Program

I gave a talk last week in the Geometry, Analysis and Physics seminar with the title “The limit operator symbol”.  This was an attempt to distill some of the ideas from my series of posts on the Lindner-Seidel and Spakula-Willett papers, especially post IV of the series.  In particular, I wanted to explain the crucial move from having a series of inequalities witnessed to having a similar series of inequalities centrally witnessed.  As Nigel put it during the seminar, we are attempting to describe a “witness (re)location program”: our witnesses are scattered all over \(\Gamma\), and we are attempting to move them all to the “courthouse”, that is, to a neighborhood of the identity, at the same time. Continue reading

Thermodynamics III: second law

The first law of thermodynamics says that heat is a form of energy. There is a lot of heat about!  For instance, the amount of heat energy it would take to change the temperature of the world’s oceans by one degree is about \(6 \times 10^{24}\) joules.  That is four orders of magnitude greater than the world’s annual energy consumption!  So, if we could somehow how to figure out how to extract one degree’s worth of heat energy from the oceans, we could power the world for ten thousand years!  Continue reading

Property A and ONL, after Kato

Hiroki Sato’s paper on the equivalence of property A and  operator norm localization was recently published in Crelle ( “Property A and the Operator Norm Localization Property for Discrete Metric Spaces.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 2014 (690): 207–16. doi:10.1515/crelle-2012-0065.) and I wanted to write up my understanding of this result.  It completes a circle of proofs that various forms of “coarse amenability” are equivalent to one another, thus underlining the significance and naturalness of the “property A” idea that Guoliang came up with twenty years ago. Continue reading

Contract signed for “Winding Around”

So I signed the contract last week for “Winding Around”, my book based on the course I taught in the MASS geometry/topology track last year.  It will appear in the American Mathematical Society’s Student Mathematical Library series, and the manuscript is due to be delivered to them on April 1st – I leave it to you whether or not you think this is an auspicious day!   The book centers around the notion of “winding number” (hence “Winding Around”) and uses that as a peg on which to hang a variety of topics in geometry, topology and analysis — finishing up, in the final chapter, with the Bott periodicity theorem considered as one possible high-dimensional generalization of the winding number notion.

The intended audience is an undergraduate one (there was skepticism from some of the AMS readers about this, but I told them the MASS students made it through okay!) and the tone is, I hope, entertaining and discursive.  As I say in the introduction, “Winding around is a description of the book’s methodology as well as of its subject-matter.”