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

Marc Rieffel 75th birthday proceedings

RieffelI just received a message about the publication of a set of conference proceedings honoring Marc Rieffel’s 75th birthday.  The proceedings originate from a program at the Fields Institute that was held last year (June, 2013).

To quote the web site of the proceedings: This special issue is a tribute to Marc A. Rieffel, marking (approximately) his 75th birthday. It is the outgrowth of a Focus Program on Noncommutative Geometry and Quantum Groups that took place at the Fields Institute for Research in Mathematical Sciences in Toronto, Canada, in June, 2013. Marc Rieffel has been one of the most influential researchers in the world in the areas of noncommutative geometry and quantum groups. He has had over 30 PhD students and over 80 “mathematical descendants”. Among his major contributions were the introduction of Morita equivalence as a fundamental notion in noncommutative geometry and the classification of C*-algebras, the introduction of stable rank as a basic invariant of C*-algebras, the introduction of strict deformation quantization to construct new examples of quantum groups, and the analysis of the metric structure of noncommutative geometries. The papers in this special issue reflect the wide range of his contributions to mathematics as well as the great esteem in which he is held by the world mathematical community.

The proceedings may be downloaded here.

Photo is copyright Mathematische Forschungsintitut Oberwolfach and licensed under Creative Commons.

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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