Category Archives: Free Probability Theory 2018/19

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 …

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.


Merry Christmas and a Happy New Year

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)