My interest in random matrix theory is two-fold. On one hand, it provides us with a very succesful model in statistical mechanics, with precise notions of universality and a rich internal mathematical structure unifying integrable systems, operator theory, orthogonal polynomials, probability theory and representation theory. On the other hand, random matrix theory meshes beautifully with numerical linear algebra. Some of the best constructions of matrix ensembles were by numerical linear algebraists. Further, random matrix theory provides a way to quantify the performance of several workhorse algorithms of applied mathematics, including techniques for the solution of linear systems, eigenvalue problems, and linear and semidefinite programming.

My goal is to tie these ends together to provide precise answers to some of the central questions in applied mathematics: what is the structure and performance of the fundamental algorithms of applied mathematics. The kind of question in this area that keeps me up at night is simple to state: what are the basic bounds on the average case and worst case performance Gaussian elimination with partial pivoting? Its been almost seventy years since Von Neumann and his co-workers began to look at such questions, but we still don't know the answer.

The explorations below began with Christian Pfrang's thesis and have now become a thriving industry that has mainly been pursued by Percy Deift and Tom Trogdon. I view the asymptotic performance of algorithms as a beautiful problem in statistical mechanics and have many problems for Ph.D students in the area. The rise of deep learning has also meant that the study of completely integrable and gradient flows on spaces of matrices provide useful benchmarks for the optimization under the hood in deep learning.

Papers on random matrix theory and numerical linear algebra

Smoothed analysis for the conjugate gradient algorithm. With Tom Trogdon, SIGMA 2016.

On the condition number of the critically-scaled Laguerre Unitary Ensemble. With Percy Deift and Tom Trogdon, DCDS-A (special volume for Peter Lax's 90th birthday), Vol. 36, (2016) .

The Airy function is a Fredholm determinant Journal of dynamics and differential equations, Vol. 28, (2016).

There is an embarrassing error in this paper and it really should be titled "The Airy function is almost a Fredholm determinant". The main result in this paper is a consequence of a recent paper by Fritz Gesztesy and Klaus Kirsten titled "on traces and modified Fredholm determinants for half-line Schrodinger operators with purely discrete spectra" (QAM, online ISSN 1552-4485).

Universality in numerical computations with random data With Percy Deift, Sheehan Olver and Tom Trogdon. PNAS (Sept. 2014).

Numerical solution of Dyson Brownian motion and a sampling scheme for invariant matrix ensembles. with Helen Li . J. Stat. Phys. Vol 153. (2013).

How long does it take to compute the eigenvalues of a random symmetric matrix? With Christian Pfrang and Percy Deift. In Random matrix theory, interacting particle systems, and integrable systems, MSRI Publications, Vol. 65. (2014).

This article is (mostly) an abbreviated presentation of extensive numerical experiments in Christian's thesis. Send me email if you'd also like a copy of his dissertation.

Lecture notes

Random matrix theory April 2018.

These notes are now part of a book in development with Tom Trogdon.

Translation of Biane's paper on the Riemann zeta function and probability theory . This is a beautiful paper on a mysterious aspect of random matrix theory.