Noncommutative Analysis

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Re: the blackboard vs. presentation debate

I came back from a couple of conferences not long ago. Here is something non-mathematical that I learned, which changed my opinion regarding the title. What I learned can be described by the following exact sequence:

0 \rightarrow I \rightarrow T \rightarrow C \rightarrow 0

Here, I represents an Israeli speaker, T represents the talk, into which the Israeli speaker injects all his knowledge into, and C represents a mostly Chinese audience, onto which the talk is mapped surjectively. Unfortunately, the kernel of the map from T to C is precisely the image of the map from I to T, so really all that the audience is left with at the end is everything in the talk modulo what the speaker was trying to say.

Paul Erdos is known for saying that the international language of Mathematics is broken English. It is true that the broken English spoken by a Hungarian, a Russian, or an Israeli are almost the same language. But there are other countries where a very different dialect of broken English is spoken. An Israeli breaks English in directions orthogonal to the way a Chinese would.

Corollary: The next time that I go to China (if they ever invite me again!!) I will prepare a presentation.

Besides the fact that I gave a talk that I thought was incomprehensible to many people, the conference was very interesting, and I met great people, and received the finest hospitality I ever did.

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.

Mathematics on mathematics

 

This post is the outline of a talk (or perhaps the talk is an outline of this post) that I will give on February 28 on our “open day” to prospective students in our department. This is supposed to be a story, it is intended to give a flavor, and neither the history nor the math are 100% precise, because it is a 15 minute talk! The big challenge is to take some rough ideas from this post, throw away the rest, and make that into an interesting quarter of an hour lecture. Comments are very welcome.

1. Introduction

What is mathematics? I am not going to answer that. You have all met mathematics in your life, in school, but also in other places as well, because math is everywhere. So you have some kind of idea what mathematics is about. However, I suspect that the most profound aspects of math have been hidden from you. I am here today to try to give you a taste of this mathematics which you have not yet seen. It is only fair to let you know that — for better and for worse — the mathematician’s mathematics, the mathematics that you will study if you do an undergraduate degree in math, is of a dramatically different nature from the math you learn in high-school or the math-is-everywhere kind of math which you meet in various popular accounts.

You have met various different kinds of mathematics: combinatorics, geometry, algebra, integral and differential calculus (aka HEDVA — which literally means “joy” in Hebrew). It seems as if mathematics splits into various branches, where in each branch there are different tasks that one should do. The objects of study of geometry are triangles, circles, trapezoids, etc; one has to prove that a certain triangle has this or that property, or one has to compute some angle or length or area. The objects of study in algebra are certain symbolic expressions or equations; one has to find the root of an equation, or to simplify an expression. In HEDVA the objects of study are functions; one has to compute the minimum of a function, or its anti-derivative, and so on.

The theme of this talk is that the objects of study in mathematics do not have to be only triangles or functions or equations, but they can also be geometry or analysis or algebra. Mathematics can also be used to study mathematics itself. This is profound. But perhaps more surprisingly, this has practical consequences.

Of course, I have no time to tell you precisely how this works. For this, I recommend that you come here and study mathematics. Read the rest of this entry »

Partial results

What is more wonderful: something wonderful, or the moment before the wonderful something happens?

Recently I proved a result that can best be described as a partial result. This is certainly not what I planned to obtain. My goal was to prove C. The plan was as follows:

  1. Prove A
  2. Prove B
  3. Prove that A and B together imply C

Step 3 is easy and I had it from the start. I’ve been trying to prove A and B for a several months now. I was sure that step 1 is the easier part and that step 2 is harder. I was quite hoping that step 1 follows from a general theorem which I also wanted to prove, but would have been just as happy to find it in the books.

Last week somebody showed me a counter example to the general theorem that would have implied A. Two days later I proved B. Now it remains to prove A, but A is not going to be true for the reasons I thought it would. Have to try a different approach…

But wait! I don’t want to prove A yet (not that I believe that there is a real “danger” in that happening). I want to enjoy B. B is beautiful. If I prove A, then C follows, and B is a triviality compared to C.

Worse, if A proves to be impenetrable, then as far as C goes, B is useless.

The role of B as a step in the proof of C can be appreciated a little more if we wait. Even if A is never proved, I wish to take some time enjoy B.

Alright, time out over! Back to A.