Geometric matrix Brownian motion is the solution (in N×N matrices) to the stochastic differential equation dGt=GtdZt, G0=I, where Zt is a Ginibre Brownian motion (all independent complex Brownian motion entries). It can also be described as the standard Brownian motion on the Lie group GL(N,C). For N>2, with probability 1 it is not a normal matrix for any t>0. Over the last 5 years, we have made progress in understanding its asymptotic moments and fluctuations, but the non-normality (and lack of explicit symmetry) has made understanding its large-N limit empirical eigenvalue distribution quite challenging. The tools around the circular law are now rich and provide a (log) potential course of action to understand the eigenvalues. There are two sides to this problem in general, both quite difficult: proving that the empirical law of eigenvalues converges (which amounts to certain tightness conditions on singular values), and computing what it converges {\em to}. In the case of the geometric matrix Brownian motion, the question of convergence is still a work in progress; but in recent joint work with Bruce Driver and Brian Hall, we have explicitly calculated the limit empirical eigenvalue distribution. It has an analytic density with a nice polar decomposition, supported on a region that resembles a lima bean for small t>0, then folds over and becomes a topological annulus when t>4. Our methods blend stochastic analysis, complex analysis, and PDE, and approach the log potential in a new way that we hope will be useful in a wider context.