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

# “Winding Around” is now available

I was excited to receive a package from the American Mathematical Society today! My author’s copies of “Winding Around” have arrived!

This is a book based on by MASS 2013 course of the same title, which looks at the *winding number* – the central notion in plane topology – from a variety of perspectives, topological, geometrical, analytic and combinatorial. Having (I hope) made the case that the winding number concept is the “golden cord which guides the student through the labyrinth of classical mathematics”, I conclude by following a beautiful paper of Michael Atiyah to explain how, by asking one natural question about the winding number, we can be led to the Bott periodicity theorem, a central result in the flowering of topology in the 1960s.

If you are interested in the book, please visit the AMS bookstore page.

I will attempt to maintain a list of typos and other corrections here. At present this list is empty, but I doubt if that happy state of affairs will last for long! (**edit:** it didn’t) Please contact me with any information about corrections. And, to quote the final sentence of the book, “I wish you much happy winding around in the future.”

# Traces and commutators

The following is a true (and well-known) theorem: \(\newcommand{\Tr}{\mathop{\rm Tr}}\)

*Suppose \(A\) and \(B\) are bounded operators on a Hilbert space, and \(AB\) and \(BA\) are trace class. Then \( \Tr(AB)=\Tr(BA) \).*

This is easy to prove if one of the operators \(A,B\) is itself of trace class, or if they both are Hilbert-Schmidt (the obvious calculation works). In the general case it is a bit harder. The “usual” argument proceeds via Lidskii’s trace theorem – the trace of any trace-class operator is the sum of the eigenvalues – together with the purely algebraic fact that the nonzero eigenvalues of \(AB\) and \(BA\) are the same (including multiplicities). Continue reading

# Proofs and Understanding

A year and a half ago I wrote a post on my ideas about using “structured proofs” to improve understanding in the Analysis I course.

I duly tried this approach, and felt that it made some difference, though perhaps not as much as I had hoped.

Now in this month’s *Notices of the AMS* there is a fascinating article by a team of educators from Loughborough University in the UK. Entitled *Understanding and Improving Undergraduate Proof Comprehension*, the article discusses as three-stage effort to help undergraduate students gain a deeper understanding of proofs – the same issue that my “structured proof” software was intended to address.

# 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**.