Math 8660: Random Matrix Theory

Spring 2014

Welcome to the course webpage!

Course Instructor: Arnab Sen
Office: 238 Vincent Hall
Email: arnab@math.umn.edu

Class time: MW 3:35 P.M. -- 4:50 P.M.
Location: Vincent Hall 207
Office hours: After lecture or by appointment.

Course Description: This course is an introduction to random matrix theory. Over the last two decades, the random matrix theory has grown into a huge field with diverse themes and has found connections to various branches of mathematics. In this one-semester course, we will attempt to cover only a small selection of topics. Our main focus will be to understand the spectrum of a large random self-adjoint matrix on both local and global scales. However, we will touch upon random non-self-adjoint matrices as well. We will explain various classical models of random matrices including Wigner matrices, Gaussian unitary ensembles, Gaussian beta ensembles, and Ginibre ensembles. In the last part of the course, we will study some spectral properties of the adjacency matrices of large sparse random graphs.

Course Content: We plan to cover the following topics:

Prerequisite: No prior knowledge in random matrix theory is required but students should be comfortable with linear algebra and basic probability theory.

Evaluation: 2 homework assignments + a final report (a short (no more than 3 pages) summary of one of the selected articles on random matrices - the list to be given later) .

This course will not have an official textbook. But the following references will be useful:

1. G. Anderson, A. Guionnet and O. Zeitouni: An Introduction to Random Matrices (available online).
2. A. Guionnet: Large random matrices: lectures on macroscopic asymptotics. Lectures from the 36th Probability Summer School held in Saint-Flour, 2006 (available online).
3. Terence Tao's blog posts on random matrices (also avaialble as a book).
4. Charles Bordenave, Djalil Chafaï: Around the circular law.
5. J. B. Hough, M. Krishnapur, Y. Peres and B. Virag: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. (available online).

Topics covered:

Homework problems.

Suggested Reading: