Georgia Institute of Technology College of Sciences

Random matrices arise naturally in various contexts ranging from theoretical physics to computer science. In a large part of these problems, it is important to know the behavior of the spectral characteristics of a random matrix of a large but fixed size. We will discuss a recent progress in this area illustrating it by problems coming from combinatorics and computer science: Condition number of “full” and sparse random matrices. Consider a system of linear equations Ax = b where the right hand side is known only approximately. In the process of solving this system, the error in vector b gets magnified by the condition number of the matrix A. A conjecture of von Neumann that with high probability, the condition number of an n × n random matrix with independent entries is O(n) has been proven several years ago. We will discuss this result as well as the possibility of its extension to sparse matrices. Random matrices in combinatorics. A perfect matching in a graph with an even number of vertices is a pairing of vertices connected by edges of the graph. Evaluating or even estimating the number of perfect matchings in a given graph deterministically may be computationally expensive. We will discuss an application of the random matrix theory to estimating the number of perfect matchings in a de- terministic graph. Random matrices and traffic jams. Adding another highway to an existing highway system may lead to worse traffic jams. This phenomenon known as Braess’ paradox is still lacking a rigorous mathematical explanation. It was recently explained for a toy model, and the explanation is based on the properties of the eigenvectors of random matrices.

We shall consider a three-dimensional Quantum Field Theory model of an electron bound to a Coulomb impurity in a polar crystal and exposed to a homogeneous magnetic field of strength B > 0. Using an argument of Frank and Geisinger [Commun. Math. Phys. 338, 1-29 (2015)] we can see that as B → ∞ the ground- state energy is described by a one-dimensional minimization problem with a delta- function potential. Our contribution is to extend this description also to the ground- state wave function: we shall see that as B → ∞ its electron density in the direction of the magnetic field converges to the minimizer of the one-dimensional problem. Moreover, the minimizer can be evaluated explicitly.

Cotlar’s identity provides an easy (maybe the easiest) argument for the Lp boundedness of Hilbert transforms. E. Ricard and I discovered a more flexible version of this identity, in the recent study of the boundedness of Hilbert transforms on the free groups. In this talk, I will try to introduce this version of Cotlar’s identity and the Lp Fourier multipliers on free groups.