Speaker: Hijoon Chae(Hongik University)
Time: May 28 (Tue), 2019 15:00-16:15.
Place: 110N104
Title: Random matrices and free probability: an introduction

This is an introductory lecture on subject of the title, requiring only a little
knowledge of calculus, linear algebra and probability at undergraduate level.

Random matrices first appeared in statistics in 1920's and become popular
since Wigner observed in 1950's that the distribution of spaces between consecutive energy levels of
heavy nuclei is similar to that of eigenvalues of large random Hermitian matrices.
Now random matrix theory has many interesting applications
(or at least inspires developments)
in diverse fields as combinatorics, algebraic geometry and number theory.

Focusing on Hermitian matrices (for simplicity), we will lecture on
eigenvalue distribution of random matrices and related topics. Also we will show that
a formal matrix integral gives a generating function of some combinatorial objects, planar maps.

Free probability is a recent development (of 1990's) which can be viewed as
a theory of non-commutative random variables. Here the notion of independence
is not obvious and is motivated by the theory of random matrices.
The ubiquity of random matrices, the semi-circle law, in particular, can be
explained as the non-commutative central limit theorem in free probability.
We will explain these topics as far as time permits.