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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Artin’s Criterion

Picture of Artin

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

\[ \int_\Gamma f(z)dz = 0. \]

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The Big Question About Limit Operators II

In the first post in this series, I gave some background to the “Big Question” on limit operators which it appears that Lindner and Seidel have solved for the case of free abelian groups.  In the next couple of posts I want to sketch some of the key ideas of their proof and to explore to what extent it can also be generalized to all exact groups (in the same way that I generalized the basic theory of limit operators to all exact groups in my 2005 paper).

There are two components to the L-S argument, it seems to me.

  • a localization property for the “lower norm” of a finite propagation operators, and
  • a “condensation of singularities” argument.

In this post we’ll look at the first of those. Continue reading

The Big Question about Limit Operators I

A recent paper on the ArXiv (see bibliography below) is entitled “An Affirmative Answer to the Big Question about Limit Operators”.   I want to do a series of posts about this paper.  In this one I will try to explain the background, at least in the most elementary (Hilbert space) case.  In this introduction I will begin by re-expressing matters in the language of coarse geometry, following my paper (also in the bibliography below).

The basic setting is a discrete group \(\Gamma\) with a (left-invariant) word metric; in the usual literature about limit operators this group is \(\mathbb Z\) or \({\mathbb Z}^n\), but there is no particular need for this restriction.  Let \(A\) denote the rough algebra of \(\Gamma\), that is the C*-algebra of operators on \(H=\ell^2(\Gamma)\) generated by \(\ell^\infty(\Gamma)\) together with the right translations by elements of \(\Gamma\); equivalently, the closure of the bounded finite propagation operators on \(H\).  If \(T\in A\), then all the translates \(L_\gamma T L_\gamma^*\) also belong to \(A\), and indeed it is easy to see that they form a precompact subset of \(A\) in the strong (or *-strong) operator topology.  The set of *-strong limit points of this subset is called the \emph{operator spectrum} of \(A\) and denoted \(\sigma_{op}(A) \). Continue reading

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?