# Proof of Harer-Zagier now online

This blog started actually a couple of years ago as a (not very successful) discussion forum for the recordings of my lectures on Free Probability, Non-Commutative Distributions, Random Matrices, Mathematical Aspects of Quantum Mechanics. During the last year I have still been producing quite a few videos, but those are on mathematics for engineers (and also in German), so not of much interest in this context here. But the Christmas break gave me some motivation and energy to do something more elaborate. So I came back to the random matrix class, where one lecture was, by some technical reasons, not recorded during the original class two years ago. I did a retake of this missing class, and so finally the playlist for the lecture series on random matrices is now also complete. The topic of this new recording is the proof of the theorem of Harer-Zagier. It gives more or less the original proof from the paper by Harer and Zagier, which I find still fun and amazing – consisting of a mixture of analysis, combinatorics, and generatingfunctionology, which I am most fond of. If you want to have a look see here, here, here, and here. For videos without audience (as is now common during corona times) I have become used to splitting the whole lecture into smaller units — which also has the advantage that I can clean the black board in between.

Maybe I will also find time and energy in a (probably far far away) future to do a reshooting of the whole free probability course. This is of course closest to my heart, but as this was my first attempt on recording lectures I needed some time to become aware of the importance of the audio quality – which is so bad in those videos that they did not make it to youtoube …

# 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 …

# 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.

# “Non-Commutative Distributions”, Summer Term 2019

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.

# Update on “Free Probability” Class

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.

# News on Lectures on “Free Probability Theory”

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.

# Lectures on “Free Probability Theory”

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.