R A N D O M    M A T R I X    T H E O R Y,    S P R I N G  2 0 1 5

Lectures: Tuesday, 1.25pm-3.15pm, in Warren Weaver Hall 1302.

Lecturer: Paul Bourgade. For office hours, you can set up an appointment or just drop by (Warren Weaver Hall 603).

Course description: This course will introduce techniques to understand the spectrum and eigenvectors of large random self-adjoint matrices, on both global and local scales. Topics include Gaussian ensembles, Dyson's Brownian motion, determinantal processes, bulk and edge scaling limits, universality for random matrices.

Prerequisites: Basic knowledge of linear algebra, probability theory and stochastic calculus is required.

Textbooks: There is no reference book for this course. Possible useful texts are:

Greg Anderson, Alice Guionnet and Ofer Zeitouni. An Introduction to Random Matrices.
Percy Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach.
Percy Deift and Dimitri Gioev, Random Matrix Theory: Invariant Ensembles and Universality.
Laszlo Erdos, lecture notes on universality for random matrices and log-gases.

Homework: Once a month.

Grading: Based on problem sets and attendance/participation at lectures.

A tentative schedule for this course is:

• Jan. 27. Blizzard.
• Feb. 3. Gaussian ensembles, eigenvalues distribution.
• Feb. 10. Gaussian ensembles, large deviations and the semicircle law.
• Feb. 17. Gaussian ensembles, local fluctuations.
• Feb. 24. Gaussian ensembles, local asymptotics.
• Mar. 3. The semicircle law, via the Stieltjes transform method.
• Mar. 10. The semicircle law, intermediate scale.
• Mar. 17. Spring break.
• Mar. 24. The semicircle law, optimal scale.
• Mar. 31. The Dyson Brownian motion.
• Apr. 7. Universality for Wigner matrices, invariance under noise addition.
• Apr. 14. Universality for Wigner matrices, Holder regularity I.
• Apr. 21. Universality for Wigner matrices, Holder regularity II.
• Apr. 23. Universality for Wigner matrices, eigenvectors.
• May 12 Open problems in random matrix theory.

• Problem set 1.
• Problem set 2.