abstract: Consider the pistil of a flower waiting to catch a grain of pollen, a lymphocyte waiting to be stimulated by an antigen to produce antibodies, or an anteater randomly foraging for an ant nest to plunder. Each of these problems can be modeled as a diffusive process with a mix of reflecting and absorbing boundary conditions. One can characterize the agent (pollen, antigen, anteater) finding its target (pistil, lymphocyte, ant nest) as a first passage time (FPT) problem for the distribution of the time when a particle executing a random walk is absorbed. In this talk we will examine a hierarchy of FPT problems modeling planar or spherical surfaces with a distribution of circular absorbing traps. We will describe a Kinetic Monte Carlo method that exploits exact solutions to accelerate a particle-based simulation of the capture time. A notable advantage of these methods is that run time is independent of how far from the traps one begins. We compare our results with asymptotic approximations of the FPT distribution for particles that start far from the traps. Our goal is to validate the efficacy of homogenizing the surface boundary conditions, replacing the reflecting (Neumann) and absorbing (Dirichlet) boundary conditions with a mixed (Robin) boundary condition.