# 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

# 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.

# 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

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

# Math 312 and “structured proving”

This coming semester I will be teaching a couple of sections of Math 312, which is the introductory real analysis course at Penn State.  The only prerequisite for this course is Calculus II (Math 141) and, in particular, students are not required to have taken an “introduction to proofs” course; though, in practice, many of them will have done so.

I have long thought that in teaching a first or second proof-based course, especially in analysis with lots of quantifiers floating about, one should try to emphasize the “block structured” nature of proofs, analogous to the block-structured nature of a programming language like C.  I had the impression that I came up with this idea for myself, but Dan Velleman wrote a whole beautiful book (How to Prove It) from this perspective, and I know I read the first edition of that book when I was in Oxford, so probably that is where I became aware of this point of view.

Anyhow, I spent a day writing some TeX macros to format “block structured” proofs.  Follow this link for a few examples.

One of the pleasing things about structuring proofs this way is that one can describe how a proof is constructed, by compressing the lower-level data.  Here for example are compressed versions of the three proofs above.

The symbol $$\require{AMSsymbols}\blacktriangle\quad\blacksquare\quad\blacktriangledown$$ for the omitted material is supposed to remind students that there are three ways to look: “up” for the givens at this point in the proof, “down” for the goals for which this part of the proof is reaching, and “across” to construct some argument linking the local givens to the local goals.

Does anyone have experience using this sort of explicitly structured proof in Analysis I? How did it work out?

# Mathematics for Sustainability website

Some readers will know that I have been thinking for a year or so about a course on Mathematics for Sustainability to teach at PSU early next year.  I’ve written regularly about this at my Points of Inflection blog, for instance here, here and here.

The official proposal for the course is wending its way through the Faculty Senate approval process.  Assuming all goes according to plan, the course will be on the books this fall, and I will offer it for the first time in Fall 2014.

I’ve created a website for this course-to-be at http://sites.psu.edu/mathforsust/.  It’s pretty much of a skeleton right now, but I plan to fill it out over the year so that it becomes a full-fledged resource for the course by next year.  So check back regularly.