# Coarse Index Theory Lecture 2

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!

# Coarse Index Theory Lecture 1

I gave the first of the two coarse index theory lectures yesterday.  The Polycom equipment makes a recording as standard, and I have uploaded it to YouTube.   So, you can take a look.  Is this an effective way to communicate mathematics?  It seemed to me to work pretty well.

I reviewed the basic definitions of the coarse index and then presented the always-elegant example of the partitioned manifold index theorem.  It seemed as though the presentation could be followed well enough by the German audience; only the business of asking and answering questions was a bit clunky.  Here is a direct link to the slides.

# 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.