AMS Open Math Notes

I wonder if you know about the AMS Open Math Notes project? I only just heard of it, but I feel very positive about any projects that make mathematical content more freely available.  Here is the link to the AMS page about the project:

As my students know, I’ve made a habit over the years of putting together TeXed notes for the courses I deliver – especially graduate courses – and now I have quite a number of them.  With some pressure on my time (read: cancer) there is no way that I could bring all of these to formal publication, even if that was the right route for them.  But as “MathNotes” I can see that they might be helpful.  So I’m going to start submitting them, perhaps after light revision, to the AMS site.  I made a start today by posting my notes from the Penn State complex analysis (graduate) course, which I’ve delivered three or four times, taking a slightly different tack each time.  Based on what I learn from that, I have a good queue of other notes to submit as well.  This is actually quite exciting for me.

Updated, December 20th: The notes have now appeared on the OpenMathNotes site, and may be found here.

In further exciting (to me) AMS news, my Winding Around made it to their 2016 bestseller list!  Because of this, the AMS is offering a special discount for orders placed between now and the end of January…

I hope to upload further packages of notes in the new year! Best wishes to all!

Artin’s Criterion

Emil Artin, picture from Wikipedia

There’s been a long lull in updates to this page.  I’ve posted elsewhere about the reason for that: in brief, I was found to have cancer, and subsequent treatments have kept me busy for months.  You can read about this on my personal website, but I’m not going to write more about it here.

Meanwhile though I have been slowly writing up a book-length version of my MASS 2013 course, “Winding Around”, whose central theme is “The Winding Number in Topology, Geometry and Analysis”.  As I was “winding around” myself and trying to complete Chapter 5 in a way that was satisfactory, I ran into an interesting “gap” in my own understanding.  This is related to the homology version of Cauchy’s theorem.  This is usually stated in the following way.

Theorem  Let $$f$$ be a function that is holomorphic on an open subset $$\Omega\subseteq\mathbb C$$, and let $$\Gamma$$ be a cycle in $$\Omega$$ that is nullhomologous, this being defined to mean that the winding number of $$\Gamma$$ about each point of $${\mathbb C}\setminus\Omega$$ is zero.   Then

$\int_\Gamma f(z)dz = 0.$