Noncommutative Analysis

Souvenirs from the children’s room, and the warmest recommendation for an online mini-course

Ilia Binder, Damir Kinzebulatov and Javad Mashreghi have organized a Focus Program on Analytic Function Spaces and their Applications at the Fields Institute, and this week, as part of this focus program, there was a Mini-course and Workshop on Drury-Arveson Space which I virtually attended (from the “children’s room” in our house, because that’s where we have the internet connection). The workshop is still not over, I have Ken Davidson’s talk to look forward to tonight.

I used to have a section in my blog Souvenirs from … where I would write about my favorite talks that I heard in recent conferences. This exercise helped concentrate during conferences (“hmmm, I wonder who’s going to be my souvenir?”) and also helped me get the most out of great talks after the conference (writing about stuff forces you to actually look up the paper or at least have another look at the notes you took during the talk). In fact, some of the souvenirs I brought home from conferences ended up becoming major parts of my own research program.

“Coming back” from the workshop on Drury-Arveson space, I can report that all the talks are recorded and can be found on the Fields Institute’s Youtube channel. To a certain extent that makes the task of reporting from conferences seem less needed.

Still, I will share my recommendations. And I want to give one very very warm recommendation for the Mini-course that Michael Hartz gave on the Drury-Arveson space. I have been to several minicourses in my life, and I also gave a couple, and I think that I have never seen a better prepared or more motivating mini-course. It was artful! Really, anybody going to work on Drury-Arveson space and the related operator theory should see it.

Here are the talks:

First lecture:

Mini-course on Drury-Arveson space, Lecture 1

Second Lecture:

Mini-course on Drury-Arveson space, Lecture 2

Third lecture:

Mini-course on Drury-Arveson space, Lecture 3

Since it was recorded, I can also put up here a link to my own talk at the workshop “Quotients of the Drury-Arveson space and their classification in terms of complex geometry”:

von Neumann’s inequality for row contractive matrix tuples

Update: we showed that the constants C_{d,n} are uniformly bounded in d for a fixed n.
Here is a link to the second version:
https://arxiv.org/abs/2109.08550

Noncommutative Analysis

Michael Hartz, Stefan Richter and I recently uploaded our paper von Neumann’s inequality for row contractive matrix tuples to the arxiv.

The main result is the following.

We prove that for all $latex d,nin mathbb{N}$, there exists a constant $latex C_{d,n}$ such that for every row contraction $latex T$ consisting of $latex d$ commuting $latex n times n$ matrices and every polynomial $latex p$, the following inequality holds:

$latex  |p(T)| le C_{d,n} sup_{z in mathbb{B}_d} |p(z)|$ .

We then apply this result and the considerations involved in the proof to several open problems from the literature. I won’t go into that because I think that the abstract and introduction do a good job of explaining what we do in the paper. In this post I will write about how this collaboration with Michael and Stefan started, and give some heuristic explanation why our result is not trivial.

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von Neumann’s inequality for row contractive matrix tuples

Michael Hartz, Stefan Richter and I recently uploaded our paper von Neumann’s inequality for row contractive matrix tuples to the arxiv.

The main result is the following.

We prove that for all d,n\in \mathbb{N}, there exists a constant C_{d,n} such that for every row contraction T consisting of d commuting n \times n matrices and every polynomial p, the following inequality holds:

 \|p(T)\| \le C_{d,n} \sup_{z \in \mathbb{B}_d} |p(z)| .

We then apply this result and the considerations involved in the proof to several open problems from the literature. I won’t go into that because I think that the abstract and introduction do a good job of explaining what we do in the paper. In this post I will write about how this collaboration with Michael and Stefan started, and give some heuristic explanation why our result is not trivial.

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My talk at BIRS on “Noncommutative convexity, a la Davidson and Kennedy”

Update August 5: here is the link to the video recording of the talk: link.

I was invited to speak in the BIRS workshop Multivariable Operator Theory and Function Spaces in Several Variables. Surprise: the organizers asked each of the invited speakers (with the exception of some early career researchers, I think) to speak on somebody else’s work. I think that this is a very nice idea for two reasons.

First, it is very healthy to encourage researchers to open their eyes and look around, instead of concentrating always on their own work – either racing for another publication or “selling” it. At the very least being asked to speak about somebody else’s work, it is guaranteed that I will learn something new in the workshop!

The second reason why I think that this is a very welcome idea is maybe a bit deeper. Every mathematician works to solve their favorite problems or develop their theories, but every once in a while it is worthwhile to stop and think: what do we make out of all this? What are the results/theories/points of view that we would like to carry forward with us? The tree can’t grow in all directions with no checks – we need to prune it. We need to bridge the gap between the never stopping flow of papers and results, on one side, and the textbooks of the future, on the other side.

With these ambitious thoughts in mind, I chose to speak about Davidson and Kennedy’s paper “Noncommutative Choquet theory” in order to force myself to digest and internalize what looked to me to be an important paper from the moment it came out, and with this I hoped to stop a moment and rearrange my mental grip on noncommutative function theory and noncommutative convexity.

The theory developed by Davidson and Kennedy and its precursors were inspired to a very large extent by classical Choquet theory. It therefore seems that to understand it properly, as well as to understand the reasoning behind some of the definitions and approaches, one needs to be familiar with this theory. So one possible natural way to start to describe Davidson and Kennedy’s theory is by recalling the classical theory that it generalizes.

But I didn’t want to explain it in this way, because that is the way that Davidson and Kennedy’s exposition (both in the papers and in some talks that I saw) goes. I wanted to start with the noncommutative point of view from the outset. I did use the classical (i.e. commutative case) for a tiny bit of motivation but in a somewhat different way, which rests on stuff everybody knows. So, I did a little expository experiment, and if you think it blew up then everybody can simply go and read the original paper.

Here are my “slides”:

The conference webpage will have video recordings of all talks at some point.

Why a “scientific approach” to science education is something I reject

Our Department has a new Teaching Seminar (concerned with teaching mathematics at the university level) which is led by legendary math professor Aviv Censor. The first lecture that I attended this semester was given by Professor Emeritus Avinoam Kolodny (Hebrew abstract here. A link to the talk – works only for Technion accounts – here). In the compelling lecture Kolodny started by mentioning the assumptions that we make when teaching (students come to class, they listen, they understand what we say, they then go home and solve homework problems) and contrasts this with empiric reality (a huge portion of students don’t come to class, the ones that do don’t listen, the ones that do don’t understand, and then they go and copy homework or solve routine problems like robots). Prof. Kolodny – an esteemed and decorated lecturer – said that he was troubled and puzzled by his students’ lack of success, and that at some point he became aware of the paper “Why not try a scientific approach to science education?” by eminent physicist, educationist and Nobel Prize laureate Carl Wieman. Kolodny explained various ideas of how to improve science (or engineering) education at the university level, to a large extent in line with ideas presented in Wieman’s paper.

The bottom line of Kolodny’s talk and Wieman’s paper is that the university lecture as we know it doesn’t work and is a waste of time. They have some ideas how to fix it, an approach that – as a first approximation – we can call “technology driven flipped classroom”. To me, the most disturbing parts of their approach are (1) that they believe that their opinions are “science based”, and therefore (2) they believe in promoting institutional change. These two aspects worry more than any technical discussion whether we should flip the classroom sideways or upside-down.

Kolodny remarked during his talk (I am paraphrasing): “I am not here to bury the concept of a lecture. Lectures are good and important. In fact, I am giving a lecture at this very moment. But you should remember that lectures are no good at passing information. In a lecture you motivate, you stimulate, you do propaganda. I’m here to do propaganda”.

Certainly I was stimulated by the talk, I was motivated to look up and then read Wieman’s paper, but most of all I was angry, I felt that someone was trying to brainwash me to believe in a certain ideology, rather than sharing some insights on teaching. Part of what made me feel this way was the “scientific approach” rhetoric. Another thing that bothered me was the jump from facts (some problems that almost everybody will agree on) to conclusions (a particular pedagogical methodology is the only way that works), disregarding tradition as not much more than momentum. Indeed, it felt like propaganda.

In this post I want to record my thoughts on some arguments raised by flipped classroom enthusiasts, and in particular on two aspects: the “scientific approach” approach, and with it the claim that lectures don’t work and we have to revolutionize the whole structure of courses to make them work.

I wish to recommend reading Wieman’s paper. Not only so that you can appreciate my criticism, but because it is a well reasoned piece of work by someone who has not only thought deeply about, but also researched the subject. I have a lot of respect for his efforts.

I am focusing my criticism on his paper, because it is written and available and interesting. But I am really arguing with talks, lectures, discussions, blog posts etc. that I have seen through the years, and have got me thinking for a long time. Now is just an opportunity to pour all of this out.

So, why not try a scientific approach to science education? Here’s why not:

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