## Four Lectures in Random Matrix Theory

**Speaker(s): ** Yiting Li (KTH)

**Time: **April 24 - April 27, 2019

**Venue: ** Room 77201, Jingchunyuan 78, BICMR

**Date: ****April 24,26,27 15:00-17:00 ****Venue: Room 78301，Jingchunyuan 78**

**Date: April 25 15:00-17:00 Venue: Room**** 29, Quan Zhai**

This mini course consists of four lectures.

In the first lecture I will introduce some basic models and well known facts in random matrix theory, such as the classic GOE, GUE and GSE models, the semicircle law, the Selberg integral and so on. I will also talk about some important results in random matrix theory such as the circular law and the Tracy-Widom Distribution for the largest eigenvalue of Wigner matrix.

In the second and the third lectures I will talk about the beta ensemble. This model is a generalization of the GOE/GUE/GSE models. I will introduce the universality property of beta ensemble proved by Bourgade, Erdos and Yau. Instead of reproducing their proof of the theorem, I will talk about some important methods they used in the proof such as the loop equation, the Sjostrand-Helffer formula and the large deviation of the empirical measure. Finally I will talk about a particular problem for beta ensemble which can also be considered as a toy model of the Riemann-Hilbert problem.

In the last lecture I plan to talk about the random matrix model A+UBU^* where A, B are deterministic real diagonal matrices and U is a random Unitary matrix under the Haar measure. I will first talk about the connection between this model and the free probability theory. Then I will introduce some recent results of this model.

Both undergraduate and graduate students are welcome. The preliminaries we need will be no more than complex variable, undergraduate probability theory and a little measure theory.