Abstract: Gaps (or spacings) between consecutive eigenvalues of a random hermitian matrix plays a central role in random matrix theory. When Wigner used random matrix as a model in physics, it was the behavior of the gaps that captured his attention. Another famous example is the Dyson-Montgomery discussion connecting these gaps with gaps between zeroes of the zeta function.

On the other hand, we still do not understand these gaps very well at the microscopic level. For instance, it has only been proved very recently by Tao and the speaker that for a random +-1 matrix, with high probability all gaps are positive.

In this talk, we will discuss some recent progress in bounding these gaps, based on purely combinatorial techniques, for very general classes of random matrices. We will also present applications in both computer science and numerical analysis (among others, solutions to questions by Babai, and by Deker-Lee-Linial).


Joint work with H. Nguyen (OSU) and T. Tao (UCLA).

Coffee will be served at 12:00 before the lecture
at Schreiber building 006