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
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.
Writing up the Connes-Renault notes, which I mentioned in a previous post, leads to a number of interesting digressions. For instance, the notion of “holomorphic closure” is discussed at some length in these early notes. But what exactly is the relationship between “holomorphic closure”, “inverse closure”, “complete holomorphic closure” (= holomorphic closure when tensored with any matrix algebra), and so on? I was aware that there had been some progress in this area but had not really sorted it out in my mind. Here’s a summary (all these results are pretty old, so perhaps everyone knows this but me…)
In 1980, Alain Connes gave a course entitled “C*-algebras, foliations and K-theory”. Jean Renault was a student in the course at that time and took notes, and photocopies of his meticulously handwritten manuscript have been passed around generations of students. I must have acquired mine some time around 1988.
The notes describe projective modules, Morita equivalence, K-theory, non-unital algebras and multipliers, quasi-isomorphisms, smooth subalgebras and “holomorphic closure”, Bott periodicity, crossed products, the Thom isomorphism for crossed products, and the beginnings of noncommutative geometry. Its fascinating to see how early some of these ideas were germinating, and what they looked like at that early stage.
In our seminar last year we assigned graduate students to read and lecture on various parts of the manuscript. This led to a rough English translation, which I’m now in the process of tidying up. I hope to post a more polished version to the arXiv before too long. I’m grateful to Alain and Jean for encouraging this project.
You can find scans (not too legible) of the lecture notes here.