Noncommutative Analysis

The complex matrix cube problem (in “Summer Projects in Mathematics at the Technion”)

Next week I will participate as a mentor in the Technion’s Summer Projects in Mathematics. The project I offered is called “Numerical explorations of open problems from operator theory”, and it suggests three open problems in operator theory where theoretical progress seems to be stuck, and for which I believe that some computer experiments can help us get a feeling of what is going on. I also hope that thinking seriously about designing experiments can help us to understand some general facets of the theory.

I have been in contact with the students in the last few weeks and we decided to concentrate on “the matrix cube problem”. On Sunday, when the week begins, I will need to present the background to the project to all participants of this week, and I have seven minutes (!!) for this. As everybody knows, the shorter the presentation, the harder the task is, and the more preparation and thought it requires. So I will take use this blog to practice my little talk.

Introduction to the matrix cube problem

This project is in the theory of operator spaces. My purpose is to give you some kind of flavour of what the theory is about, and what we will do this week to contribute to our understanding of this theory.

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The day I got tenure

I was on the phone, and there was a knock on my door. I mumbled something and in came the dean. “Oh, I see that you’re busy, I’ll come back later.”

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Polya’s three rules of style

In G. Polya‘s book “How to Solve It”, one of the shortest sections is called “Rules of style”. This section contains Polya’s three rules of style, which are worth repeating.

“The first rule of style”, writes Polya, “is to have something to say”.

“The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time”.

Polya’s third rule of style is: “Don’t say what does not need to be said” or maybe “Don’t say the obvious”. I am not sure of the exact formulation, because Polya doesn’t write the third rule down – that would be a violation of the rule!

Polya’s three rules are excellent and one is advised to follow them if one strives for good style when writing mathematics. However, style is not the only criterion by which we measure mathematical writing. There is a tradeoff between succinct and elegant style, on the one hand, and clarity and precision, on the other.

“Don’t say the obvious” – sure! But what is obvious? And to whom? A careful writer leaving a well placed exercise in a textbook is one thing. An author of a long and technical paper that leaves an exercise to the poor, overworked referee, is something different. And, of course, a mathematician leaving cryptic notes to his four-months-older self, is the most annoying of them all.

“Don’t say the obvious” – sure, sure! But is it even true? I think that all the mistakes that I am responsible for publishing have originated by an omission of an “obvious” argument. I won’t speak about actual mistakes made by others, but I do have the feeling that some people have gotten away with not explaining something non-trivial, and were lucky that things turned out to be as their intuition suggested (granted, having the correct intuition is also a non-trivial achievement).

I disagree with Polya’s third rule of style. And you see, to reject it, I had to formulate it. QED.

Souvenirs from the Red River

Last week I attended the annual Canadian Operator Symposium, better known in its nickname: COSY. This conference happens every year and travels between Canadian universities, and this time it was held in the University of Manitoba, in Winnipeg. It was organized by Raphaël Clouâtre and Nina Zorboska, who altogether did a great job.

My first discovery: Winnipeg is not that bad! In fact I loved it. Example: here is the view from the window of my room in the university residence:

20180604_053844

Not bad, right? A very beautiful sight to wake up to in the morning. (I got the impression, that Winnipeg is nothing to look forward to, from Canadians. People of the world: don’t listen to Canadians when they say something bad about any place that just doesn’t quite live up to the standard of Montreal, Vancouver, or Banff.) Here is what you see if you look from the other side of the building:  Read the rest of this entry »

The perfect Nullstellensatz

Question: to what extent can we recover a polynomial from its zeros?

Our goal in this post is to give several answers to this question and its generalisations. In order to obtain elegant answers, we work over the complex field \mathbb{C} (e.g., there are many polynomials, such as  x^{2n} +1, that have no real zeros; the fact that they don’t have real zeros tells us something about these polynomials, but there is no way to “recover” these polynomials from their non-existing zeros). We will write \mathbb{C}[z] for the algebra of polynomials in one complex variable with complex coefficients, and consider it as a function of the complex variable z \in \mathbb{C}. We will also write \mathbb{C}[z_1, \ldots, z_d] for the algebra of polynomials in d (commuting) variables, and think of it – at least initially – as a function of the variable z = (z_1, \ldots, z_d) \in \mathbb{C}^dRead the rest of this entry »

Minimal and maximal matrix convex sets

The final version of the paper Minimal and maximal matrix convex sets, written by Ben Passer, Baruch Solel and myself, has recently appeared online. The publisher (Elsevier) sent us a link through which the official final version is downloadable, for anyone who clicks on the following link before May 26, 2018. Here is the link for the use of the public:

Click here to download the journal version of the paper

Of course, if you don’t click by May 26 – don’t panic! We always put our papers on the arXiv, and here is the link to that. Here is the abstract:

Abstract. For every convex body K \subseteq \mathbb{R}^d, there is a minimal matrix convex set \mathcal{W}^{min}(K), and a maximal matrix convex set \mathcal{W}^{max}(K), which have K as their ground level. We aim to find the optimal constant \theta(K) such that \mathcal{W}^{max}(K) \subseteq \theta(K) \cdot \mathcal{W}^{min}(K). For example, if \overline{\mathbb{B}}_{p,d} is the unit ball in \mathbb{R}^d with the p-norm, then we find that 

\theta(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p-1/2|} .

This constant is sharp, and it is new for all p \neq 2. Moreover, for some sets K we find a minimal set L for which \mathcal{W}^{max}(K) \subseteq \mathcal{W}^{min}(L). In particular, we obtain that a convex body K satisfies \mathcal{W}^{max}(K) = \mathcal{W}^{min}(K) only if K is a simplex.

These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every d-tuple of self-adjoint contractions, can be dilated to a commuting family of self-adjoints, each of norm at most \sqrt{d}. We also introduce new explicit constructions of these (and other) dilations.

Ronald G. Douglas (1938-2018)

A couple of weeks ago I learned from an American colleague that Ron Douglas passed away. This loss saddens me very much. Ron Douglas was a leader in the Operator Theory community, an inspiring mathematician, a person of the kind that they don’t make like any more.

The first time that I met him was in the summer of 2009, Read the rest of this entry »

Souvenirs from San Diego

Every time that I fly to a conference, I think about the airport puzzle that I once read in Terry Tao’s blog. Suppose that you are trying to get quickly from point A to point B in an airport, and that part of the way has moving walkways, and part of it doesn’t. Suppose that you can either walk or run, but you can only run for a certain small amount of the time. Where is it better to spend that amount of time running: on the moving walkways or in between the moving walkways? Does it matter?

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The nightmare

In September 30 the mathematician Vladimir Voevodsky passed away. Voevodsky, a Fields medalist, is a mathematician of whom I barely heard earlier, but after bumping into an obituary I was drawn to read about him and about his career. His story is remarkable in many ways. Voevodsky comes out as brilliant, intellectually honest giant, who bravely and honestly confronted the crisis that he observed “higher dimensional mathematics” was in.

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I quit (from MathSciNet and ZbMath)

(This is the post that I wanted to write this weekend.)

Several months ago I informed both MathSciNet as well as Zentralblatt that I would like to stop reviewing papers for these repositories. If you don’t know what I am talking about (your PhD thesis advisor should be fired!), then MathSciNet and Zentralblatt are databases that index published papers in mathematics, contains some bibliographic information (such as a reference list for every paper, as well as a list of papers that reference it), and, significantly, has a review for every indexed paper. The reviews are written by mathematicians who do so voluntarily (they get AMS points or something). If the editors find nobody willing to review, then the abstract appears instead of a review. This used to a very valuable tool, and is still quite valuable.

I quit because:

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