Course on “Non-Commutative Distributions”, summer term 2019

  • the lectures are videotaped, the recordings can be found here
  • for blog entries related to this class check this link
  • the homepage of the course can be found here

hand-written lecture notes

Section 0: Introduction

Section 1: Basic Definitions and Examples

Section 2: Operator-Valued Distribution and Operator-Valued Cauchy Transform

 Section 3: Non-Commutative Functions: Motivations and Formal Aspects

Section 4: The Operator-Valued Cauchy Transform

Section 5: Operator-Valued Freeness

Section 6 (part 1, part 2): Operator-Valued Central Limit Theorem and Operator-Valued Semicircular Elements

Section 7: Matrices of Semicirculars and Matrix-Valued Semicirculars (and Block Random Matrices)

Section 8: Polynomials in Free Semicirculars and Linearization

Section 9: Combinatorial and Analytic Description of Operator-Valued Freeness: Free Cumulants and R-Transform

Section10: Operator-Valued Free Convolution via Subordination Function and the Distribution of Polynomials in Free Variables

Computer slides from the end of Section 10 (beginning of Lecture 19)

Section 11: Distribution of Rational Expressions in Free Random Variables

Section 12: Unbounded Rational Expressions