Noncommutative Analysis

Month: July, 2013

A survey on the Drury-Arveson space, and a new paper

I just finished writing (and re-writing) a survey paper on the Drury-Arveson space. Here it is. I am posting a link to this survey in hope that it will be useful for somebody (I also need to figure in what open online form I am going to keep this article). I will also be very happy to hear comments about mistakes or glaring omissions I made. I am planning to submit this survey to a handbook in operator theory project, and there are some places in the text such as :”SEE CHAPTER ON DILATION THEORY” which is supposed to refer to other chapters in the handbook.

This is the first time that I am writing such a comprehensive survey article, and it is much, much harder than I thought it would be. The hardest part is to find the balance of which references to give and which references not to give. A survey as it is cannot contain all information, and I think that a good survey should make it clear to the reader which are the most important references. That’s a big responsibility!

I wrote a series of blog posts on Drury-Arveson space: one, two and three. Actually, at the time I wrote those posts I already knew that I was going to write this survey, and it was kind of a warm up.

I take this opportunity also to put up a link to this a new paper By Michael Hartz, Ken Davidson and myself, “Multipliers of embedded discs”. In this paper we continue our journey to understand the algebraic structure of complete Pick algebras in terms of the varieties on which they naturally live, I explained this problem in this older post

Course announcement: Function Theory in Several Complex Variables

Here is an interesting quote that comes to my mind again and again:

You teach best what you most need to learn. (Richard Bach)

This coming winter I will be giving a course in function theory in several complex variables. As last year, I may use the blog to post some lecture notes for lectures that I will present in a somewhat different way than how it is done in the books I have. I don’t know yet the course number. The course will be between two to four hours a week. Probably it will be four.

Here is a very interesting introduction to the subject, by R. Michael Range, that gives a flavor of in what ways this subject is much more than a straightforward marriage of function theory in one complex variables and multi-variable calculus.

My incentives for offering this course are

  1. First of all, my need to use complex analysis in several variables has been steadily growing since the last year of my PhD studies, and I want to fasten all the knowledge I accumulated, get it organize in my head, and dive deeper.
  2. I am thinking of this course as part of the training that I owe my students and postdocs. These days working with me means, with high probability, doing something related to function theory in several complex variables.
  3. This is a beautiful, beautiful subject, which is very important to other fields such as algebraic geometry and some fancy physics (but now that we marked off “importance” we can safely forget about it — at least for now — and concentrate on “beautiful”), and should be given in our department from time to time, ve im lo achshav, eimatai?  

This course will not be designed to ultimately reach my current research interests, because there are so many basic and classical material that I want to cover which come before my current interests, and are by far more important (and probably more interesting, to most students) than what I happen to be doing right now.