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Epsilon propagation and the Roe algebra

I remember with great pleasure giving the CBMS lectures that became the book Index Theory, Coarse Geometry and the Topology of Manifolds. I was at the beginning of a six month visiting appointment, delighting in the change to be with my family in this beautiful city, and energized by coming up with two lectures every day (and I mostly wrote the book as we went along – how nice to be young and energetic).

However this speedy process meant that one (or maybe more) over-optimistic statements slipped by. One in particular that has caused trouble over the years is the Remark after Lemma 3.5  In this Remark I define a notion of \(\epsilon\)-propagation (notation being as in the usual setup for Roe algebras):

Definition: The \(\epsilon\)-propagation of an operator \(T\) is the infimal \(R\) such that

\[ \|f\|\le 1, \ \|g\|\le 1, \ d(\text{Support}(f),\text{Support}(g))\le R \Longrightarrow  \|fTg\| \le \epsilon. \]

It is clear that an operator in the Roe algebra has finite \(\epsilon\)-propagation for all \(\epsilon\). It is also clear (though I didn’t point this out at the time) that the collection of operators having finite \(\epsilon\)-propagation for all \(\epsilon\), which some people call quasi-local operators, is a \(C^*\)-algebra. So the question arises: is it the same as the Roe algebra?

So here’s where my optimism came in. I causally remarked, “If \(X\) is large scale finite dimensional—by which I meant what we would now call finite asymptotic dimension—then the converse is the case”  (“the converse” being the statement that every quasi-local operator is in the Roe algebra). I thought that there would be a fairly obvious “large scale patrition of unity” proof—and somehow never checked. There isn’t.  Mea culpa.  Over the years people proved the statement for \(\mathbb Z\) and then \({\mathbb Z}^n\), but these proofs used Fourier series and clearly didn’t generalize.

Until this year.  Back last summer I received a note from Aaron Tikuisis of Aberdeen asking about the status of the statement and saying that he had a proof.  This has now led to a preprint by Tikuisis and Spakula showing the equivalence not just for spaces of finite asymptotic dimension but also of “finite decomposition complexity” (see Guentner, Tessera and Yu, A notion of geometric complexity and its application to topological rigidity, Inventiones 189 (1012), 315-37).  They say it will be on the arXiv soon.  I just had another inquiry from a grad student about this misleading remark of mine and was happy, this time, to be able to point him to an honest proof.

If only I’d called it a “conjecture”!  Still, I am very happy that the question is resolved in such a nice way.  The paper uses some of the new ideas in classification, like nuclear dimension, though I have not read it in enough detail to be able to explain how.

 

 

Michael Atiyah’s Birthday!

Heads up!  In  a couple of days (April 22nd) it is the 87th birthday of “Britain’s mathematical pope”, (not just Britain’s, either, IMO), otherwise known as my doctoral advisor, Professor Sir Michael Atiyah.   HAPPY BIRTHDAY MICHAEL!

To celebrate, his son David is assembling an online tribute – see http://www.atiyah.eu/mfa87/    Please consider sending a tribute message to david@atiyah.eu  Here’s what hes ays:

We are collecting messages of congratulations on the occasion of Michael Atiyah‘s 87th birthday Friday, April 22, 2016.

If you have the time, memory, and an inclination, please also include your favourite personal story about Britain’s Mathematical Pope*. I keep hearing every mathematician has one – it would be a shame not to collect and archive them for posterity.

Bonus points awarded for photographs, with prizes for the best MP4 video message we can share on the night.

Pls include:
– your name
– your current position, & location (if appropriate)
– when and where you first met Michael

We will keep it simple and hope to collate and publish submisssions in due course.

* = with thanks to Siobhan Roberts for the expression used in her recent biog of J H Conway – i have simply extended his Popedom from England to Britain.

If you haven’t seen it, here is a great article from Wired last week: Mathematical Matchmaker Atiyah Dreams of a Quantum Union.

 

Jost Bürgi’s Method for Calculating Sines

Jost Bürgi (28.2.1552 -31.1.1632) Astronom, Mathematiker, Instrumentenbauer, Entdecker der Logarithmen. aus:7523 (Rar);Frontispiz

Jost Bürgi (28.2.1552 -31.1.1632)
Astronom, Mathematiker, Instrumentenbauer, Entdecker der Logarithmen.
aus:7523 (Rar);Frontispiz

I just learned (via Facebook, no less) of a fascinating paper with the above title by Andreas Thom and coauthors.

Jost Burgi (1552-1632) was a Swiss mathematician, astronomer and clockmaker.  He worked with Johannes Kepler from 1604 and is thought to have arrived at the notion of logarithms independent of Napier.  He was also reputed to have constructed a table of sines by a brand new method, but until now the details of his Kunstweg (“artful method”) for computing sines were thought to have been lost.  The beautiful book of van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry,  (Princeton University Press, 2009) documents how computing trigonometric ratios was  a central theoretical and practical preoccupation of ancient mathematics. We know of Burgi’s Kunstweg via a statement of his colleague and friend Nicolaus Ursus: “the calculation (of a table of sines)… can be done by a special way, by dividing a right angle into as many parts as one wants; and this is arithmetically. This has been found by Justus Burgi from Switzerland, the skilful technician of His Serene Highness, the Prince of Hesse.”  But the details are not clear and apparently nobody, starting with Kepler himself, was ever able to reconstruct Burgi’s method.

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Thermodynamics IV: entropy

In the previous post, I talked about the second law of thermodynamics: there can be do thermodynamic transformation whose overall effect is to move heat from a cooler body to a hotter one.  Since the reverse of such a transformation (moving heat from a hotter body to a cooler one) happens naturally by conduction, the second law naturally contains an element of irreversibility which it is natural to expect is expressed by an inequality.   The quantity to which this applies is the famous entropy.

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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