As we gain access to more datasets with finer resolution, what assumptions should we make in our models? Idealized models in population genetics, which often lack spatial structure and other realistic features of populations, are well studied and allow for easy and quick simulation of data such as genetic sequences from a sample. However, there is a huge gap between mathematical analysis and robust statistical inference for models that consider realistic features, partly due to a lack of rigorous understanding of these models. Gaining a mathematical understanding of well-designed models is essential both for explaining the datasets at hand and for gaining insight into the complex systems under study. For example, understanding the relationships between a population’s dynamics and its genealogies is crucial to determining what is robust against model assumptions and what is not.

In this talk, I will first focus on recent applications of coalescent models to large human genomics datasets. We developed a sampling theory that not only explains dramatic differences in the site frequency spectra (SFS) across the human genomes, but also enables estimation of the mutation rates and the numbers, ages and sizes of recurrent mutations. Our method is robust against, or insensitive to, details of the population dynamics and weak selection. I then highlight the key role and challenges of understanding models with spatial structure and their potential applications to virus co-infection spread. Finally, I feature some future research directions in relation to making optimal use of the growing amount of data.