# Higher index theory with change of fundamental group

I gave a talk in our seminar yesterday which arises from trying to understand the paper of Chang, Weinberger and Yu (Chang, Stanley, Shmuel Weinberger, and Guoliang Yu. “Positive Scalar Curvature and a New Index Theorem for Noncompact Manifolds,” 2013) where they use relative index theory in a non $$\pi-\pi$$ situation to produce examples of manifolds with strange positive-scalar-curvature properties (e.g., a  non-compact manifold which has an exhaustion by compact manifolds with boundary carrying nice positive-scalar-curvature metrics, but which itself carries no such metric).

I wanted to develop an approach to this kind of index theory that was more accessible (to me) and the talk was a report on my efforts in that direction.  Here are the slides from that talk.

# The Witness Relocation Program

I gave a talk last week in the Geometry, Analysis and Physics seminar with the title “The limit operator symbol”.  This was an attempt to distill some of the ideas from my series of posts on the Lindner-Seidel and Spakula-Willett papers, especially post IV of the series.  In particular, I wanted to explain the crucial move from having a series of inequalities witnessed to having a similar series of inequalities centrally witnessed.  As Nigel put it during the seminar, we are attempting to describe a “witness (re)location program”: our witnesses are scattered all over $$\Gamma$$, and we are attempting to move them all to the “courthouse”, that is, to a neighborhood of the identity, at the same time. Continue reading

# What is noncommutative geometry?

I’m giving an introductory seminar talk this afternoon to let new graduate students know about the noncommutative geometry research group at Penn State and what it is we do.  My plan is to begin with a short “elevator speech” about NCG (a few minutes) and then follow it up with four ten-minute vignettes of “Things we talk about a lot”

• Hilbert space
• K-theory
• Curvature
• Expanders

and at least to indicate the existence of all of the $$(4 \times 3)/2 = 6$$ connections among these concepts as well.  Here is a link to a scanned version of my notes for the talk.

# My talk at BIRS

I gave a talk yesterday (August 8th, 2013) on Ghostbusting and property A.  Thanks to the technology system at BIRS you can watch the talk on video here.

The paper has now been accepted for the Journal of Functional Analysis.