Tag Archives: exposition

Metric approach to limit operators

In a couple of posts earlier this year (post I and post II) I started getting to grips with the paper An Affirmative Answer to the Big Question on Limit Operators by Lindner and Seidel. The first of these posts gives some background to limit operator theory and to what the big question is that Lindner and Seidel solved for the case of the group  {\mathbb Z}^n .  Circumstances prevented me continuing the post series, but I suggested that the arguments should work just as well for any group whose underlying metric space has property A (that is, for any exact group).

Now, I am reading the paper A metric approach to limit operators by Jan Spakula and Rufus WIllett, in which they carry out this kind of idea in much greater generality than I had been imagining.   Following the limit operator literature, they don’t simply confine their attention to the Hilbert space as I did; their arguments work on \( \ell^p(X;E) \) with \( 1 < p < \infty \) and coefficients in an auxiliary Banach space \(E\).   What’s more, their notion of limit operator does not even require an underlying group structure (and so the Fredholm theory that they develop will work for all bounded geometry discrete metric spaces that have property A).   In this post I want to explain their generalized definition of “limit operator”, as a preliminary to getting into the analysis proper. Continue reading

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

Surgery for Amateurs

In 1996 I was the Ulam Visiting Professor at the University of Colorado, Boulder.  While I was there I gave a series of graduate lectures on high-dimensional manifold theory, which I whimsically titled Surgery for Amateurs.

The title was supposed to express that I was coming to the subject from outside – basically, trying to answer to my own satisfaction the question “What is this Novikov Conjecture you keep talking about?” Perhaps because of their amateurish nature, though, these lectures struck a chord, and I have received many requests for reprints of the lecture notes.  In 2004 I began a project of revising them with the help of Andrew Ranicki; but, alas, other parts of life intervened, and the proposed book never got finished.

Obviously some people still value the material, and my plan is to try and republish it in blog form, along with comments and discussion.  The Surgery for Amateurs blog is now live and your participation is welcomed!