The Benjamin-Ono equation is a completely integrable model for weakly nonlinear internal waves that has a nonlocal dispersion term reflecting the nonlocal nature of the Dirichlet-Neumann mapping for the velocity potential at the free surface. It is both physically and mathematically interesting to consider the dynamics of the Benjamin-Ono equation in the limit when the coefficient of dispersion is small. This talk will describe an approach to the small dispersion limit based on the analysis of a certain determinantal tau function.  The main result of this analysis is a weak convergence theorem that describes the limiting behavior in terms of the branches of the multivalued solution of the inviscid Burgers equation that arises from the Benjamin-Ono equation upon setting the coefficient of dispersion to zero.  We will describe this result and give some outline of its proof, which is based on a deterministic analogue of Wigner's moment method from random matrix theory.  This talk describes joint work with Zhengjie Xu.