“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.”

Topology and music

A new paper appeared on the arXiv this week applying some simple topological ideas to the analysis of musical modes.  This contains some simple ideas about fundamental groups and so on and I was thinking it might make a project for a musically-oriented student in my MASS course.  There have been several papers of this sort recently: the earliest that I’m aware of is Rachel Hall’s article in Science five years ago. (Dr Hall is a professor at Saint Joseph’s University in Philadelphia and before that was a student in the Geometric Functional Analysis group at Penn State.)

A mode is a seven-note scale.  The familiar major scale of Western music (C,D,E,F,G,A,B) is a mode (Ionian mode), but one can obtain different modes by starting from a different note (e.g. Aeolian mode, A,B,C,D,E,F,G) or by using a different starting scale, or both. The authors analyze a mode into two chords: the seventh chord made by notes I, III, V, and VII of the scale, and the triad made by notes II, IV and VI.  For instance, Ionian mode is a major seventh (CEGB) plus a minor triad (DFA); Aeolian mode is a minor seventh (ACEG) plus a diminished triad (BDF).   Spaces of chords are natural examples of configuration spaces (selections of $$n$$ distinct points from the space of notes, with order irrelevant) so this allows the representation of a mode as a subspace of a product of two configuration spaces.

I’m not really competent to judge the music theory here (as a guitarist, I’m usually sticking with two or three of the innumerable possible scales that the authors identify), but it is a fascinating point (this is also what I got out of Rachel Hall’s paper) that music is full of natural examples of configuration spaces, including knots and Möbius bands and other topological exotica.

References

Bergomi, Mattia G., and Alessandro Portaluri. Modes in Modern Music from a Topological Viewpoint. ArXiv e-print, September 3, 2013. http://arxiv.org/abs/1309.0687.

Budney, Ryan, and William Sethares. Topology of Musical Data. ArXiv e-print, July 4, 2013. http://arxiv.org/abs/1307.1201.

Callender, Clifton, Ian Quinn, and Dmitri Tymoczko. “Generalized Voice-Leading Spaces.” Science 320, no. 5874 (April 18, 2008): 346–348. doi:10.1126/science.1153021.

Hall, Rachel Wells. “Geometrical Music Theory.” Science 320, no. 5874 (April 18, 2008): 328–329. doi:10.1126/science.1155463.

The Eisenbud–Levine–Khimshiashvili signature formula

I learned last week of a really cool result, published when I was a first-year undergraduate, that I had not been aware of before.  Maybe everyone knew it except me, but it is so neat I’m going to write about it anyway.

To set the scene, think about the Hopf index theorem for vector fields on a (compact, oriented)  $$n$$-manifold.  Continue reading

“Winding Around” now going up

The website for my MASS course, “Winding Around” (Math 497C, Fall 2013) is now live.

Winding Around” is an introduction to topology using the winding number as a unifying theme. It’s intended to be different from most introductory topology courses because we’ll try to define the key concept (winding number) as economically as possible and then  apply it in many different ways.

One of the inspirations for this course is the classic expository paper

Atiyah, M. F. “Algebraic Topology and Elliptic Operators.” Communications on Pure and Applied Mathematics 20, no. 2 (1967): 237–249. doi:10.1002/cpa.3160200202.

and if things go according to plan I hope that we may get to discuss the Bott periodicity theorem at the end of the course, in the spirit of Atiyah’s article.