Noncommutative Function Theory and Free Probability at Oberwolfach

I spent the week 28.4 – 3.5 at the MFO at Oberwolfach in a workshop on noncommutative function theory and free probability (whatever the hell that means), where I gave (ahem, ahem) a three lecture mini-course “Noncommutative Function Theory for Free Probabilists for Everyone”. It is a curious exercise to give a mini-course to a crowd that consists of about 45 superb mathematicians, about a third of which know at least as much as I do about the subject, and another fraction know almost nothing about it. It was hard work, and so I did not bring back any souvenirs.

I spent the week sadly thinking to myself, how could such a place like MFO exist? How could it be that every week a group of 48 mathematicians get pampered and fed, rest, hike, drink, give some talks and hear some talks, and all this to foster research in (usually pure) mathematics. When one raises one’s head from the scribbles on one’s notepad, and looks at the state of the world, it is hard not to think: how can this be?

I told wise old Bill Helton, one of the kind godfathers of our field, that I can’t believe that this place exists, and asked him whether with time one gets used to the idea. Does he believe that Oberwolfach exists? He answered “Of course! The world would fall apart without it.”

Arveson’s hyperrigidity conjecture refuted by Bilich and Dor-On

Boom! This morning Boris Bilich and Adam Dor-On published a short preprint on the arXiv “Arveson’s hyperrigidity conjecture is false” in which they provide a counter example that refutes Arveson’s hyperrigidity conjecture. This is a fantastic achievement! It is one of the most interesting things that happened in my field lately and also somewhat of a surprise, a paper that is sure to make a significant impact on the subject.

(I should say that Adam was kind enough to let me read the manuscript a week ago, so that I had time already to check the details and as far as I can tell it looks correct.)

Let us recall quickly what the conjecture is (for more background see the series of posts that I wrote for the topics course I gave several years ago).

Let be a unital operator algebra generating a C*-algebra .

On dilations of CP-semigroups (my talk at the Midrasha at Weizmann Institute)

I had the privilege of being invited this year again to give a talk at the Group Theory Seminar at the Weizmann Institute (Here is a link to last year’s talk). I am pasting here a link to the recording of the talk; the link is to the third minute, the recording started three minutes before the lecture began.

Title: Dilations of CP-semigroups via subproduct systems and superproduct systems of C*-correspondences.

Abstract:

The title is a bit of a mouthful, so let us unpack it together:

1. A C*-correspondence is a certain kind of bimodule over a C*-algebra B that has a B-valued inner product.
2. Sub and super-product systems are families of C*-correspondences that enjoy certain semigroup-like properties under the tensor product. 
3. A CP-semigroup is a family of completely positive maps that form a semigroup under composition; the most important examples are when the semigroup is given by a family (T_t) parameterized by positive real t>0.
4. By dilation of CP-semigroup we mean a specific way of exhibiting one CP-semigroup as a part of another semigroup that belongs to a better understood category, typically a semigroup of *-homomorphisms.

The problem of constructing dilations for CP-semigroups or determining their existence/uniqueness has drawn some of the best minds in operator algebras. And still there are open problems. I will describe the elaborate framework of sub and super-product systems employed by Michael Skeide and myself to attack this problem, a framework which is built on top of decades of work by ourselves and others. Using this framework we resolve some problems in the multi-parameter case; more surprisingly, we also obtain new results in the one-parameter case.

My teaching statment

One bright morning I got an email from the department that I need to write a teaching statement. “What is this? What is this for?” I asked (the last time that I was asked to write a teaching statement was when I applied for a postdoc in North America in fall 2008, since then I got two tenure track positions and was promoted in Israeli universities and a teaching statement was never required). They answered: You know, teaching statement. What is your vision on teaching, your teaching philosophy. Now, I am totally convinced that one’s formulation of a teaching philosophy is completely independent of one’s actual teaching performance. To spell it out: nobody needs to have a teaching philosophy, and nobody should care about someone else’s teaching philosophy. On the other hand, I do happen to have some ideas on the subject (a philosophy of teaching, if you must) so why not write them down and send them in.

Human beings have evolved to be learners and teachers. The never ending drive to see what lies beyond the next mountain, to discover how the world operates, and to invent new tools, is what defines us as humans. We know how to learn, we want to learn. We know how to teach and we love to teach. Better: we know how to learn how to learn and teach, and we are a gifted species in our ability to do this in an ever changing environment.

“CP-semigroups and dilations… and beyond” is now in print

Michael Skeide’s and my big joint paper “CP-semigroups and dilations, subproduct systems and superproduct systems: the multi parameter case and beyond” is now in print. Actually, I received the copies a couple of months ago, but I didn’t feel like posting anything. I blogged a few years back about what a difficult project this was, and how long it took to complete it. It also took quite to get published and we had to overcome some very annoying technical issues – I wrote about that here. But in the end, finally, the printed version of the paper appeared as a standalone “book” in the excellent journal Dissertationes Mathematicae (it is the analogue of Memoirs of the AMS published by the Institute of Mathematics in the Polish Academy of Sciences). Now I know that the project is really over. It feels wonderful to hold a copy in my hand. It is much more fun to read a physical copy than an electronic one.