Noncommutative Analysis

Category: Talks

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.

Slides of my talk at the seminar “in” Bucharest

This Wednesday I gave a talk at the Institute of Mathematics in Bucharest, live on zoom. Here are the slides:

In this talk, I decided to put an emphasis on telling the story of how we found ourselves working on this problem, rather than giving a logical presentation of the results in the paper that I was trying to advertise (this paper). I am not sure how much of this story one can get from the slides, but here they are.

My talk at Fields – video available

Hi just got an email from the Fields Institute that the video recording of my talk “CP-semigroups and Dilations, Subproduct Systems and Superproduct systems,: the Multiparameter Case and Beyond” (on my paper with Michael Skeide I announced here) that I gave at COSY, is now available. The slides are available here. This is the first time I see (and hear!) a recording of myself giving a talk in English and, wow, it’s devastating. Thanks for bearing with me 🙂

Here is a link to the talk:

Seminar talk at the BGU OA Seminar

This coming Thursday (July 2nd, 14:10 Israel Time) I will be giving a talk at the Ben-Gurion University Math Department’s Operator Algebras Seminar. If you are interested in a link to the Zoom please send me an email.

I will be talking mostly about these two papers of mine with co-authors: older one, newer one. Here is the title and abstract:

Title: Matrix ranges, fields, dilations and representations

Abstract: In my talk I will present several results whose unifying theme is a matrix-valued analogue of the numerical range, called the matrix range of an operator tuple. After explaining what is the matrix range and what it is good for, I will report on recent work in which we prove that there is a certain “universal” matrix range, to which the matrix ranges of a sequence of large random matrices tends to, almost surely. The key novel technical aspects of this work are the (levelwise) continuity of the matrix range of a continuous field of operators, and a certain quantitative matrix valued Hahn-Banach type separation theorem. In the last part of the talk I will explain how the (uniform) distance between matrix ranges can be interpreted equivalently as a “dilation distance”, which can be interpreted as a kind of “representation distance”. These vague ideas will be illustrated with an application: the construction of a norm continuous family of representations of the noncommutative tori (recovering a result of Haagerup-Rordam in the d=2 case and of Li Gao in the d>2 case).

Based on joint works with Malte Gerhold, Satish Pandey and Baruch Solel.

My slides for the COSY talk and the seminar talk

Here is a link to the slides for the short talk that I am giving in COSY.

This talk is a short version of the talk I gave at the Besancon Functional Analysis Seminar last week; here are the slides for that talk.