# Thermodynamics

I’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.”

# Metric approach to limit operators VI

I thought I’d finished this sequence of posts with number five, but then I spent a little time talking with Jerry Kaminker and Rufus Willett and I think that I understood two things: first, how to formulate the limit point construction more cleanly and, second, the “symmetry breaking” role of the ultrafilters which is not clear in what I had written so far.  Read on. Continue reading

# Metric approach to limit operators IV

In this post I’ll finally get to the “condensation of singularities” argument that was invented by Lindner and Seidel in the (free abelian) group context and generalized by Spakula and Willett to metric spaces.  (Calling this “condensation of singularities” is my idea, but it does seem to me to get at what is going on.  I can’t help feeling that there should be a way of replacing some of the explicit constructions with an abstract argument involving the Baire category theorem.  But I have not yet been able to come up with one.) Continue reading

# Metric approach to limit operators III

This is a continuation of my posts on the Spakula-Willett paper Metric approach to limit operators (see part I and part II).  In this post I will talk about “lower norm witnesses” on spaces with property A.  (This is quite close to what is done in my earlier post here, though using direct geometric tools rather than the functional analysis tricks I suggested, which only work in the Hilbert space case.)  Then in the next post I will talk about the “condensation of singularities” argument that completes the proof. Continue reading