Tag Archives: Dirac operator

Coarse Index Theory Lecture 2

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Here is the follow-up lecture (second of two) on coarse index theory. I tried to bear in mind that the conferees in Germany had heard quite a few presumably much more detailed presentations in between by lectures 1 and 2, so I attempted to give a fairly “big picture” overview.  I had prepared to talk about several examples that I didn’t have time to discuss, so you will find some slides at the end of the presentation below that were not talked about in the video.

Here’s the video of Lecture 2:

 

And here is the link to the corresponding slides. Hope you find the presentation helpful and enjoyable!

Dense domains, symmetric operators and spectral triples

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A cautionary paper appeared on the arXiv last week

Forsyth, I., B. Mesland, and A. Rennie. Dense Domains, Symmetric Operators and Spectral Triples. ArXiv e-print, June 6, 2013. http://arxiv.org/abs/1306.1580.

The paper is about the definition of unbounded Fredholm modules and how certain, initially plausible, weakenings of the definition do not work as expected. (Some well-known authors are called out for insufficient attention to these points; I was initially worried that Analytic K-Homology appears in the bibliography, but I don’t think Nigel and I are among the criminals here.)

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“Mass endomorphism, surgery and perturbations”

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That is the title of an interesting paper just posted on the arXiv. I had never heard of the “mass endomorphism” so this was new to me… Take a compact Riemannian spin manifold and suppose that the metric is flat in the neighborhood of a point p.  If there are no harmonic spinors (so that the Dirac operator is invertible) then the Dirac Green’s function, i.e. the inverse of the Dirac operator, has an asymptotic expansion near p in which the zero term is an endomorphism of the spinor bundle called the mass operator.  It is known that if the mass operator is non-zero then a solution exists to the classical Yamabe problem.  In this paper it is shown that the mass operator is “generically” non zero – using a lot of the machinery from positive-scalar-curvature land: psc surgery, results of Stolz, etc…