Significance
Spatial dynamics are important for understanding genetic diversity in many contexts, such as cancer and infectious diseases. Coalescent theory offers a powerful framework for interpreting and predicting patterns of genetic diversity in populations, but incorporating spatial structure into the theory has proven difficult. Here, we address this long-standing problem by studying the coalescent in a spatially expanding population. We find the topology of the coalescent depends on the growth dynamics at the front, but not on the functional form of the growth function. Instead, the transition between coalescent topologies is determined by a single dynamical parameter. Our theory makes precise predictions about the effects of population dynamics on genetic diversity at the expansion front, which we confirm in simulations.