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:
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.
I have now put up scans of my hand-written notes for the class, see here, and will update those irregularly.
The class is still running well and more or less according to plan. After generalities on non-commutative distributions, non-commutative (fully matricial) functions, and operator-valued Cauchy transforms we are now bringing some structure into our non-commutative distributions, by looking on operator-valued freeness. I plan to cover the basic part of the theory of operator-valued freeness, in particular, operator-valued additive convolution, both from a combinatorial and an analytic point of view. However, much of this is parallel to the scalar-valued theory from last term, so I will be quite brief on details (in particular, proofs) at many places – one should look back to and compare with the relevant parts from last term; in particular, Sections 2, 3, 4, 5 of the corresponding class notes.
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.