Category Archives: Teaching

“Winding Around” is now available

JohnWAI 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.

Continue reading

A correction to EOTAM

In my book Elliptic operators, topology and asymptotic methods (both the first and the second editions) I give a discussion of the representation theory of the groups Spin and Pin which was based (as far as I can now remember) on some notes that I took when I attended Adams’ famous course on the exceptional Lie groups, as a Part III student in 1981.  I no longer seem to have those, unfortunately (although meanwhile a version of Adams’ own notes on his approach has been published by University of Chicago Press).  Meanwhile, in 2010 Darij Grinberg pointed out on Math Overflow that the argument I gave was garbled: see this link.  In this post I want to explain what is garbled and how the useful part of the argument can be recovered.  Continue reading

A “well known interpolation formula”

In the Atiyah-Bott paper on their Lefschetz theorem for elliptic complexes, they give a very nice elementary example of the Lefschetz theorem for the Dolbeault complex, by considering the automorphism of   {\mathbb C}{\mathbb P}^n given by

\[ [z_0,\ldots, z_n] \mapsto [\gamma_0z_0, \ldots, \gamma_nz_n ] \]

in homogeneous coordinates, where the \(\gamma_i\) are distinct and nonzero complex numbers.  This has \( (n+1) \) simple fixed points and applying the holomorphic Lefschetz theorem gives

\[ 1 = \sum_{i=0}^n \frac{\gamma_i^n}{\prod_{j\neq i} (\gamma_i-\gamma_j)}. \]

This is Example 2 on p. 460 of the second Atiyah-Bott paper.  They go on to describe this as a “well known interpolation formula”. Continue reading