My class on “Non-commutative distributions” started today. The first lecture is already online, see our video platform. Actually, we have a new video system, so the sound should now be better than last term. I am not sure, though, whether this also applies to the frames.
The class will run during our summer term, which will end mid July. Since I will travel quite a bit during term, there will be some cancellations and reschedulings of lectures; nevertheless, I still hope that we will have in the end again something like 25 lectures.
The general topic of the class is progress which was made in the last couple of years on non-commutative distributions, and which relies on advances in
- the operator-valued version of free probability theory (in particular, for its analytic description)
- free analysis or free non-commutative function theory
- relating analytic questions about operators in von Neumann algebras with the theory (of Cohn et al.) of non-commutative linear algebra or the free skew field (aka as non-commutative rational functions)
- using the linearization trick to relate non-linear scalar problems with linear operator-valued problems
All of the above will be explained in the lectures. So don’t worry if you have no idea what all this actually means.
Much of this progress was actually achieved in recent years in the context of my ERC-Advanced Grant on “Non-Commutative Distributions”. As this grant has finished now, the class can also be seen as kind of final report for this.
I will assume some familiarity with basic functional analysis and complex analysis. It is surely also helpful to know at least a bit about free probability theory, but this can also be acquired by watching along the way a few of the videos from last term or reading relevant parts of the corresponding class notes.
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.
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.
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.