Developments in topological gravity

This note aims to provide an entree to two developments in two-dimensional topological gravity - that is, intersection theory on the moduli space of Riemann surfaces - that have not yet become well known among physicists. A little over a decade ago, Mirzakhani discovered(1,2) an elegant new proof of the formulas that result from the relationship between topological gravity and matrix models of two-dimensional gravity. Here we will give a very partial introduction to that work, which hopefully will also serve as a modest tribute to the memory of a brilliant mathematical pioneer. More recently, Pandharipande, Solomon, and Tessler(3) (with further developments in Refs. 4-6) generalized intersection theory on moduli space to the case of Riemann surfaces with boundary, leading to generalizations of the familiar KdV and Virasoro formulas. Though the existence of such a generalization appears natural from the matrix model viewpoint - it corresponds to adding vector degrees of freedom to the matrix model - constructing this generalization is not straightforward. We will give some idea of the unexpected way that the difficulties were resolved.