Tag Archives: exposition

Matt Wiersma on exotic group C*-algebras

Recently Matt Wiersma from Waterloo spoke in our seminar about some of his work related to “exotic group C*-algebras”.  A more detailed account is on the arXiv.  I thought I would try to write up some of what I learned (probably, as usual, this is the most elementary points, but it was new to me).

What is an exotic group C*-algebra?  It is a completion of the group algebra which is different from the two standard examples (maximal and reduced) that we describe in C*-algebra courses.  Oversimplifying, we might make an analogy with compactifications of a locally compact Hausdorff space.   There is always a minimal one (one-point compactification) and a maximal (Stone-Cech), but there are also plenty of other things in between.  Analogously, in the case where a group \(\Gamma\) is non amenable, one might imagine that there should be many other C*-completions of \({\mathbb C}\Gamma\) lying between the maximal and the reduced C*-algebras.   (Whether, in fact, there exists any group for which \({\mathbb C}\Gamma\) has exactly two distinct completions appears to be an open question.)

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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


ThermoI’ve been trying to learn a bit about classical thermodynamics, using Fermi’s lecture notes which are available as a low-cost Dover reprint.

That’s partly just because the subject has always been a bit mysterious to me and I would like to understand it better, but also because the Second Law of thermodynamics often gets invoked in environmental discussions – and I wonder whether it is being used accurately.  (See this blog post for extended discussion about that.)

As a mathematician, I expected the discussion of thermodynamics to be statistical, heavily engaged with probability theory.  But the main text of Fermi’s book is not about statistical mechanics at all.  Instead, it is about classical thermodynamics; the nineteenth century theory that attempted to quantify the properties of that mysterious fluid, “heat”, and its transmission from one body to another. 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.”