Tag Archives: winding number

Author interview on AMS “Book Ends”

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Eriko Hironaka was kind enough to interview me for the AMS blog “Book Ends” whose logo is above. The interview focuses on Winding Around, but I suppose appropriately enough, it winds about a bit too, from “what got you started on writing” to “do you have advice for new authors”.  I enjoyed being able to share a bit through this piece. It begins:

What made you decide to write the book “Winding Around”? The spark for Winding Around was lit when I was about nine. My dad drew an incredibly convoluted simple closed curve (something like Figure 4.3 in the book), made a dot on the paper somewhere in the midst of the convolutions, and asked me, “Is that inside or outside the curve?”

If you want to read more, the whole piece is here.

 

“Winding Around” is now available

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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.”

 

Contract signed for “Winding Around”

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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.”

 

Artin’s Criterion, Part II

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In this post I want to sketch the proof of “Artin’s criterion”, following Ahlfors’ book as referred to in my previous post (and presumably following Artin himself, though I still have not come up with any original reference to him).  The argument generalizes easily to \((n-1)\)-dimensional cycles in an open subset \(\Omega\subseteq {\mathbb R}^n\), but for simplicity I will stick to the planar case.

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The Eisenbud–Levine–Khimshiashvili signature formula

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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 →