﻿ 尹晟——AN INTRODUCTION TO FREE PROBABILITY AND RANDOM MATRICES

# 尹晟——AN INTRODUCTION TO FREE PROBABILITY AND RANDOM MATRICES

Free probability theory is a quite young mathematical theory initiated by Voiculescu during 1980s in the theory of operator algebras. It provides a probabilistic point of view to investigate questions on operator algebras. Its core concept is the so-called free independence, which models the relation of free groups and parallels the notion of independence in classical probability theory.

Later on, free independence was revealed to be more than an non-commutative analogue of the classical concept of independence by a surprising connection (which was also discovered by Voiculescu) with the world of random matrices. Roughly speaking, free independence describes the asympototic behavior of a large class of independent random matrices as the dimension tends towards infinity. This connection led to very fruitful interactions that benefit both free probability and random matrices, as well as many other mathematical theories and even physics and engineering.

This course will serve as an introduction to free probability and random matrices. In particular, we will look at both sides to present the structure of freeness via combinatorial, analytical and probabilistic notions and tools. We will first cover the basics of free probability and random matrices as well as the fundamental relation between these two worlds. More advanced topics that are related to some recent research might be discussed after if the schedule allows.

Prerequisites are the basic undergraduate courses on basic analyis, linear algebra, measure theory etc. Having an operator algebra course is helpful, but not required as we will recall the relevant basic knowledge for the audience.

(1) A. Nica, R. Speicher: Lectures on the Combinatorics of Free Probability. Cambridge University Press, 2006

(2) J. Mingo, R. Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.

(3) D. Voiculescu, K. Dykema, A. Nica: Free random variables. CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992

(4) T. Tao: Topics in random matrix theory. Graduate Studies in Mathematics, vol.132.

(5) G. Anderson, A. Guionnet, O. Zeitouni: An introduction to random matrices.

(6) T. Kemp: Introduction to random matrix theory. Lecture notes.