s Freie Wahrscheinlichkeit Lehre

## Non-commutative Distributions

(Summer Semester 2019)

### News

• There is an info sheet about the exam available.
• The written exam will be on Fri, 30.08.2019, at 9:00 st in SR 6 (Room 217 building E2.4).
• There are handwritten lecture notes online available

### Lecture

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: the operator-valued version of free probability theory (combinatorial but also analytic aspects); the linearization trick to reduce non-linear scalar problems to linear operator-valued problems; the combination of (i) and (ii) to calculate the distribution of polynomials in free variables; 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; free analysis, which is a version of complex analysis for several non-commuting variables; recent results concerning the regularity of functions in non-commuting variables; 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.

### Tutorials

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

Tutor Alexander Mang (alex.mang[at]t-online.de).
Mailbox 14

### Exam

There will be a written exam, the most important information can be found here:

Date: 30.08.2019
Time: 9:00 st -12:00 st
Room: SR 6 (217) in building E2.4

For more details see the info sheet.

Exercise Sheet 1
Exercise Sheet 2
Exercise Sheet 3
Exercise Sheet 4
Exercise Sheet 5
Exercise Sheet 6
Exercise Sheet 7
Exercise Sheet 8
Exercise Sheet 9

### References

• L.A. Harris: Fixed Points of Holomorphic Mappings for Domains in Banach Spaces. Abstract and Applied Analysis, 2003 link
• D. Jekel. Operator-Valued Non-Commutative Probability link
• Kaliuzhnyi-Verbovetskyi, Dmitry S., and Victor Vinnikov. Foundations of free noncommutative function theory. Mathematical Surveys and Monographs, Vol. 199. American Mathematical Soc., 2014.
• S. Krantz: The Caratheodory and Kobayashi Metrics and Applications in Complex Analysis. The American Mathematical Monthly, 2008 link
• James A. Mingo, Roland Speicher: Free Probability and Random Matrices.
Fields Institute Monographs, Vol. 35, Springer, New York, 2017.
• Roland Speicher: Free Probability Theory; and its Avatars in Representation Theory,
Random Matrices, and Operator Algebras; also Featuring: Non-commutative Distributions
.
Jahresber. Dtsch. Math. Ver. (2017) 119: 3.
• Speicher, Roland. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Vol. 627. Memoir of American Mathematical Soc., 1998. link

updated: 7 June 2019   Tobias Mai