Conditional Coalescence Theory in a Population of Varying Size

Imagine taking a sample of genes from a certain population. If we trace these genes back in time through past generations up to their most recent common ancestor, what does the resulting tree look like? 

Coalescent theory is the mathematical study of questions of this type. The bulk of coalescent theory has dealt with constant population size models that give rise to the Kingman coalescent in the large population limit. However, real populations (e.g. the human population) rarely have constant size. Furthermore, for practical applications, the sample types are often already known (e.g. the blood types for a sample of individuals), so it is useful to understand the coalescent process between individuals in the sample conditional on such information. 

As many nice properties of the Kingman coalescent are lost in this generalization, our research focuses on small and limiting cases to make larger conjectures about the behavior for arbitrary sample sizes and arbitrary population sizes.