s Freie Wahrscheinlichkeit Lehre

Prof. Dr. Roland Speicher

Felix Leid

Non-commutative Distributions

(Summer Semester 2019)



Monday and Friday, 12:00 -- 13:30, Lecture Hall IV, Building E2 4

The lectures will present some of the main progress which we made in recent years concerning our investigations of non-commutative distributions of non-commuting operators or random matrices. In particular, a basic problem is to find a good approach to the meaning of “non-commutative distribution”. In particular, the lectures will cover:
  1. the operator-valued version of free probability theory (combinatorial but also analytic aspects);
  2. the linearization trick to reduce non-linear scalar problems to linear operator-valued problems;
  3. the combination of (i) and (ii) to calculate the distribution of polynomials in free variables;
  4. the basic theory of non-commutative rational functions (a.k. free skew field) and the extension of (iii) to such rational functions of free variables;
  5. free analysis, which is a version of complex analysis for several non-commuting variables;
  6. recent results concerning the regularity of functions in non-commuting variables;
  7. more cool stuff, which still has to be determined.
In some sense this is a continuation of my class “Free Probability Theory” from last term. On the other hand, I will develop the theory of free probability theory again, but in a more general, operator-valued context. So, in principle and with some additional efforts, it should be possible to take the class without having a prior knowledge on free probability. Big parts of the material do also not deal so much with free variables, but more general with analytic and algebraic aspects of maximal non-commuting variables. In any case, complex analysis and functional analysis are necessary as prerequisites.

Lecture announcement

For more details and updates see the blog of Roland Speicher.

Lecture notes

The lectures will be recorded and the videos will be uploaded on our video platform.


Date & Time: Mon 14-16
Room: 8 (318)

Tutor Alexander Mang (alex.mang[at]t-online.de).


Exercise Sheet 1
Exercise Sheet 2
Exercise Sheet 3
Exercise Sheet 4


updated: 1 April 2019   Tobias Mai