# The saga ends …

I have now finished my class on random matrices. The last lecture motivated the notion of (asymptotic) freeness from the point of view of looking on independent GUE random matrices. So you might think that there should now be continuations on free probability and alike coming soon. But actually this part of the story was already written and recorded and if you don’t want to spoil the tension you should watch the series not in its historical but in its logical order:

1. Random Matrices (videos, homepage of class)
2. Free Probability Theory (videos, homepage of class)
3. Non-commutative Distributions (and Operator-Valued free Probability Theory) (videos, homepage of class)

More information, in particular the underlying script (sometimes in a handwritten version, sometimes in a more polished texed version), can be found on the corresponding home page of the lecture series.

May freeness be with you …

# Is there an impact of a negative solution to Connes’ embedding problem on free probability?

There is an exciting new development on Connes’ embedding problem. The recent preprint MIP*=RE by Ji, Natarajan, Vidick, Wright, Yuen claims to have solved the problem to the negative via a negative answer to Tsirelson’s problem via the relation to decision problems on the class MIP* of languages that can be decided by a classical verifier interacting with multiple all powerful quantum provers. I have to say that I don’t really understand what all this is about – but in any case there is quite some excitement about this and there seems to be a good chance that Connes’ problem might have a negative solution. To get some idea about the excitement around this, you might have look on the blogs of Scott Aaronson or of Gil Kalai. At the operator front I have not yet seen much discussion, but it might be that we still have to get over our bafflement.

Anyhow, there is now a realistic chance that there are type II factors which are not embeddable and this raises the question (among many others) what this means for free probability. I was asked this by a couple of people and as I did not have a really satisfying answer I want to think a bit more seriously about this. At the moment my answer is just: Okay, we have our two different approaches to free entropy and a negative solution to Connes embedding problem means that they cannot always agree. This is because we always have for the non-microstates free entropy $\chi^*$ that $\chi^*(x_1+\sqrt\epsilon s_n,\dots,x_n+\sqrt\epsilon s_n)>-\infty$, if $s_1,\dots,s_n$ are free semicircular variables which are free from $x_1,\dots,x_n$. The same property for the microstates free entropy $\chi$, however, would imply that $x_1,\dots,x_n$ have microstates, i.e., the von Neumann algebra generated by $x_1,\dots,x_n$ is embeddable; see these notes of Shlyakhtenko.

But does this mean more then just saying that there are some von Neumann algebras for which we don’t have microstates but for which the non-microstates approach give some more interesting information, or is there more to it? I don’t know, but hopefully I will come back with more thoughts on this soon.

Of course, everybody is invited to share more information or thoughts on this!

# Welcome to the Non-Commutative World!

About two weeks ago I posted with Tobias Mai on the archive the preprint “A Note on the Free and Cyclic Differential Calculus”. Here is what we say in the abstract:

In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu’s fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. One the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples – such as a discrete version of the Ito stochastic integral – can be treated.

At the moment I am not in the mood to say more specifically about this preprint (maybe Tobias or I will do so later), but I want to take the opportunity — in particular as the first anniversary of this blog is also coming closer — to put this in a bigger context and mumble a bit about the bigger picture and our dreams … so actually about what this blog should be all about.

Free probability theory has come a long way. Whereas born in the subject of operator algebras, the realization that is also has to say quite a bit about random matrices paved the way to its use in many (and, in particular, also applied) subjects. Hence there are now also papers in statistics, like this one, or in deep learning, like this one or this one, which use tools from free probability for their problems. The last words on how far the use of free probability goes in those subjects are surely not yet spoken but I am looking forward to see more on this.

This is of course all great and nice for our subject, but on the other hand there is also a bigger picture in the background, where I would hope for some more fundamental uses of free probability.

This goes roughly like this. There is the classical world, where we are dealing with numbers and functions and everything commutes; then there is our non-commutative world, where we are dealing with operators and limits of random matrices and where on the basic level nothing commutes. That’s where quite a bit of maximal non-commutative mathematics has been (and is still being) developed from various points of views:

• free probability deals with a non-commutative notion of independence for non-commuting random variables;
• there is a version of a non-commutative differential calculus which allows to talk about derivatives in non-commutative variables; my paper with Tobias mentioned above is in this context and tries to formalize and put all this a bit further;
• free analysis (or free/non-commutative function theory) aims at a non-commutative version of classical complex analysis, i.e., a theory of analytic functions in non-commuting variables;
• free quantum groups provide the right kind of symmetries for such non-commuting variables.

The nice point is that all those subjects have their own source of motivation but it turns out that there are often relations between them which are non-commutative analogues of classical results.

So, again this is all great and nice, BUT apart from the commutative and our maximal non-commutative world there is actually the, maybe most important, quantum world. This is of course also non-commutative, but only up to some point. There operators don’t commute in general, but commutativity is replaced by some other relations, like the canonical commutation relations, and there are actually still a lot of operators which commute (for example, measurements which are at space-like positions are usually modeled by commuting operators). Because of this commutativity, basic concepts of free probability do not have a direct application there.

Here is a bit more concretely what I mean with that. In free probability we have free analogues of such basic concepts as entropy or Fisher information. There are a lot of nice statements and uses of those concepts and via random matrices they can also be seen as arising as a kind of large N limit of the corresponding classical concepts. However, in the classical world those concepts have usually also a kind of operational meaning by being the answer to fundamental questions. For example, the classical Shannon entropy is the answer to the question how much information one can transmit over classical channels. Now there are quantum channels and one can ask how much information one can transmit over them; again there are answers in terms of an entropy, but this is unfortunately not free entropy, but von Neumann entropy, a more commutative non-commutative cousin of classical entropy. There are just too many tensor products showing up in the quantum world which prevent a direct use of basic free probability concepts. But still, I am dreaming of finding some day operational meanings of free entropy and similar quantities.

Anyhow, I hope to continue to explain in this blog more of the concrete results and problems which we have in free probability and related subjects; but I just wanted to point out that there are also some bigger dreams in the background.

# Class on “Random Matrices”, Winter Term 2019/20

Our winter term has just started, running from mid October 2019 to mid February 2020, with a two-week break around Christmas. This term I am giving an introduction to random matrices. Again, the lectures will be recorded and put online. The lectures can be found on our video platform; more info on the lectures are also on the website of the class.

The lectures will follow roughly the material from the same class of summer term 2018, for which there exist also texed lecture notes. There will be a few reorganizations and shifts in the material, so there might emerge also a new version of the lectures notes sometimes in the future …

# Update on videos of lectures and talks

## Videos of talks at Fields Institute

The videos of my three talks in the distinguished lecture series at the Fields Institute are now online and can be found here.

In particular, the first talk, entitled Dan-Virgil Voiculescu: visionary operator algebraist and creator of free probability theory, was a talk for a public audience and gives not only a bit of information on Dan Voiculescu, but also a very high level idea of what free probability is all about. And it also has some movie references …

## Videos of lecture series “Non-commutative distributions”

The course on “Non-commutative distributions” has now finished; all 20 lectures are online and can be found on our video platform. My hand-written notes for the classes can be found here.

# Focus Program on Applications of Noncommutative Functions; Featuring a Celebration Banquet for Dan Voiculescu’s 70th Birthday

Today started the Focus Program on Applications of Noncommutative Functions at the Fields Institute in Toronto. There will be two workshops: this week on the “Developments and Technical Aspects of Free Noncommutative Functions” and next week on “Applications to Random Matrices and Free Probability of Free Noncommutative Functions”. Both workshops look interesting to me; unfortunately I will miss most of the first one as I will fly only on Wednesday to Toronto.

I will give a series of three talks on the relation between free probability and random matrices. The first talk will be quite general and is also intended for a public academic audience. Its main purpose is to celebrate the 70th birthday of Dan Voiculescu by giving an idea of Dan’s achievements and of free probability theory. The talk and the banquet will be on the very day of Dan’s birthday.