MATH 254A : Topics in Random Matrix Theory

• Course description: Central limit theorem, concentration of measure and random walks.  Universal laws for bulk spectral distributions (e.g. Wigner semi-circular law, Marchenko-Pastur law, Circular law).  Introduction to free probability.  Spectral properties of the Gaussian Unitary Ensemble and its relatives; relationship with orthogonal polynomials and Riemann-Hilbert problems.  Universality of spacing distributions for eigenvalues and singular values, both in the bulk and the edge of the spectrum.  Connections with the Littlewood-Offord problem.  Connections with Dyson Brownian motion and the Ornstein-Uhlenbeck process.  In the (unlikely) event that time permits, I may also discuss the conjectured connections with the statistics of the Riemann zeta function.

Announcements:

• I will not be giving classes on Mon Mar 8 and Wed Mar 10.

• Instructor: Terence Tao, tao@math.ucla.edu, x64844, MS 6183
• Lecture: MWF 10-10:50, MS5137
• Quiz section: None
• Office Hours: M 2-3
• Textbook: I will use a number of sources, including Deift's "Orthogonal polynomials and random matrices: A Riemann Hilbert approach", Deift-Gioev's "Random matrix theory: invariant ensembles and universality"; Mehta's "Random Matrices"; and Gionnet "Large Random Matrices: Lectures on Macroscopic Asymptotics".  I will post lecture notes on my blog site.
• Prerequisite: 245A and 275A are highly recommended (and 245B, 275B will also be helpful).  At a bare minimum, one should have an advanced undergraduate or higher exposure to several variable calculus, real analysis, complex analysis, linear algebra (particularly the spectral theorem), and probability theory.  Some exposure to combinatorics and the classical Lie groups (U(n) and O(n) in particular) may also be helpful.
• Grading: This is a topics course, so I am planning a fairly informal grading scheme.  Basically, the base grade will be B provided you actually show up to a significant number of classes, and adjusted upwards according to whether you turn in any homework.
• Reading Assignment: Lecture notes will be provided on my blog site.  Students are encouraged to comment on these posts.  Note that the notes will cover more material than the lectures.  You may also wish to read my survey talk slides on random matrices, which roughly correspond to the topics we will cover in this course.
• Homework: There will be five homework assignments, assigned from the lecture notes:
1. First homework (Due Fri Jan 15): Do Exercise 3 from Notes 1 (but ignore the part about comparison with Exercise 2).  Then do Exercise 6 from Notes 1.  (The case of large A is already sketched out in Proposition 6; what I want to see here is how to modify that argument to deal with the case when, say, A < 1.)