Our one-month program on New Developments in Free Probability and Applications has started. After a one-week workshop on the more theoretical side, there will be now two weeks of introductions and survey talks, as well as seminar talks, before we finish with another workshop on the (potential) applications of free probability.
Here is the schedule of talks for this week; they will be held in room 5340, 5th floor, Andre-Aisenstadt building on the campus of the Universite de Montreal.
Note that the second talk on Friday has changed; Felix Leid’s talk has been moved to next week, Ben Hayes will talk instead.
TUESDAY March 12
9:30 – 12:30 Tobias Mai, Roland Speicher, and Sheng Yin
“Introduction to Regularity and the Free Field”
2:30 – 4:00 Hari Bercovici
“Introduction to outliers”
THURSDAY March 14
9:30 – 12:30 Benson Au, Guillaume Cébron, and Camille Male
“Introduction to Traffic Freeness”
2:30 – 3:30 Laura Maassen
“Group-theoretical quantum groups”
4:00 – 5:00 Simon Schmidt
“Quantum automorphism groups of finite graphs”
FRIDAY March 15
9:30 – 10:30 Josué Váquez Becerra
“Fluctuation moments induced by some asymptotically liberating unitary matrices”
11:00 – 12:00 Ben Hayes
“A random matrix approach to the Peterson-Thom conjecture”
The lectures on our free probability class are over now. All 26 lectures are uploaded on the video platform, and the scans of my handwritten course notes can be found here.
I plan to continue next term with a class on “Non-Commutative Distributions”; this will in particular cover the operator-valued version of free probability and its use for dealing with polynomials in free variables, as well as addressing regularity properties of the distribution of such polynomials. There will be more cool stuff, but I still have to think about details. Our summer term starts in April, then I will be back with more information.
The plan is to continue with the recording of the lectures. If you have any suggestions on how to improve on this, please let me know.
When preparing my lectures for the asymptotic freeness of various random matrix ensembles I stumbled about the situation concerning Wigner matrices. We all know that Wigner matrices and deterministic matrices are asymptotically free, but all the proofs I am aware of are annoyingly complicated. Shouldn’t there be a nice and simple proof without too many technicalities?
As was said in the section “Open problems and possible future directions” of the report of the 2008 Banff workshop “Free Probability, Extensions, and Applications”: “Whereas engineers have no problems in applying asymptotic freeness results for unitarily invariant ensembles it has become apparent that they do not have the same confidence in the analogous results for Wigner matrices. The main reason for this is the lack of precise statements on this in the literature. This has to be remedied in the future.”
Actually, at that time there existed already some results in this direction in the paper On Certain Free Product Factors via an Extended Matrix Model from 1993 of Ken Dykema. There the asymptotic freeness between independent Wigner matrices and diagonal (or even more general: block diagonal) deterministic matrices had been shown. But the case of general deterministic matrices remained open. Taking on the challenge by the engineers, we were determined to write down nice and accessible proofs, also including the full deterministic case.
The result of this was that such statements and proofs were included in the book An Introduction to Random Matrices of Greg Anderson, Alice Guionnet, and Ofer Zeitouni on one side and in my book with Jamie on the other side. However, I have to admit what looked like an easy exercise to Jamie and me at the beginning turned out to me much more complicated. An intermediate outcome of this was my paper with Jamie on Sharp Bounds for Sums Associated to Graphs of Matrices, but even with this as a nice black box the final proof still required a couple of technical pages in our book.
So I would like to come back to the original challenge and want to see what we really know about the relation between Wigner matrices and deterministic matrices. What are the clean statements and what are nice proofs. The situation for Wigner matrices is also more complicated than for Gaussian matrices, as the real and imaginary part for Wigner matrices do not have to be independent, hence the complex situation cannot be directly reduced to the real one, and questions about the *-freeness of non-selfadjoint Wigner matrices is also not exactly the same as the freeness of selfadjoint ones. Of course, all is related and similar, but if one asks a concrete question, usually it is hard to find the answer exactly for this in the literature.
I hope to collect here information about what is out there in the literature on that problem, with the final goal of coming up with the cleanest statements and the simplest proofs. If you have any information or ideas in this context, please let me know.
The lectures on free probability are back!
I have now started to put scans of my handwritten lecture notes online; they correspond more or less to what I write on the blackboard. As we are now, in the context of random matrix calculations, having a lot of indices hanging around it might in some cases be easier to decipher those from my notes than from the blackboard. Maybe sometimes in the future I will even tex them, but don’t count on this … In any case, most of the material I am presenting in this class is either from my book with Andu or from my book with Jamie, so that there exist already nicely written notes on this.
There is of course much more to say about random matrices. One issue is that in this class I cover only convergence and asymptotic freeness results in the averaged sense. Of course, almost sure versions of those results usually also exist. For more on those and other aspects of random matrices I refer to the random matrix literature (part of which you can find on the homepage of my class “Random Matrices” from last term). There exists also a nice tex-ed version of the lectures notes from my class on random matrices.
2019 will again be a year with quite a few meetings around free probability.
The main event will be a month-long program on New Developments in Free Probability and Applications at CRM in Montreal in March 2019. There will be two workshops: one, at the beginning of March, on the theory and its extensions and the second, at the end of March, on the applied perspective. In the two weeks in between there will also be quite some activity, in particular, we are aiming at bringing graduate students and postdoctoral fellows quickly to the frontiers of the subject. Furthermore, Alice Guionnet will give the Aisenstadt Chair lecture series between both workshops.
This program is part of the year long celebration of the CRM’s 50th anniversary. It seems very appropriate to have such a meeting on the blossoming of free probability theory, and its promise for the future at the place where the seed was sown. In the spring of 1991 Dan Voiculescu was the holder of the Andre Aisenstadt chair at the CRM in Montreal during the ’91 operator algebra program. At this time, free probability was still in its infancy and only known to a small group of enthusiasts. This was going to change. Voiculescu gave the Aisenstadt Lectures on free probability in Montreal, organizing the material and bringing it with the help of his students Ken Dykema and Alexandru Nica into a publishable form. The resulting book was the first volume in the CRM Monograph Series and was instrumental for making the theory more generally accessible and attracting many, in particular young, researchers to the subject. It is still the most cited literature on free probability. Andu, Dan, and Ken (as well as a couple of other experts) will stay as Simons Scholars-in-Residence for the whole program at CRM
Another month-long program with a substantial free probability component will be the Focus Program on Applications of Noncommutative Functions at the Fields Institute in Toronto, June 10 – July 5, 2019. In particular, one of the workshops of the program, June 17-21, deals with applications of noncommutative functions to random matrices and free probability.
The focus program at the Fields Institute will also include a celebratory banquet on June 14, in honour of the 70th birthday of Dan Voiculescu.
This is a blog on topics around Free Probability Theory. Originally, I created this to provide a forum around my lecture on free probability theory. But now I plan (at least hope) to extend it to general blog on free probability theory.
So I hope to post here also all kind of information which is relevant in the context of free probability theory, like: meetings, new results, discussions of open problems or general directions in the subject.
I hope that others will also make some contributions; if you are interested in writing your own posts in this context, please contact me!
We have now the Christmas break for our term. This means there will be no classes for two weeks. We will continue on January 7, 2019; there will be about ten more lectures. We will then cover two more main themes:
i) the random matrix connections of free probability (in particular, show the asymptotic freeness of many random matrix models)
ii) applications of free probability to von Neumann algebras (in particular, treat results about compressions of free group factors)
This is a blog on my videotaped lectures on “Free Probability Theory” from the winter term 2018/2019 at Saarland University.
Here is the link to the recordings of the lectures.
For now, the main purpose of this blog is to give viewers the oportunity to comment on the lectures or ask questions about the content.