Tag Archives: winding number

Artin’s Criterion, Part II

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

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.