Research

I am interested in applications of algebraic geometry in biology. Currently my main area of focus is algebraic statistics for phylogenetics or the study of evolutionary relationships between taxa. A common research question is whether we can infer past evolutionary events from observable genetic data. Usually this means fitting a graphical probability model to our data. These models are used to describe some set of evolutionary events, such as how species evolved together and how long ago, they split off from common ancestors. Phylogenetic networks are used to model hybridization, horizontal gene transfer (the movement of DNA between organisms that don’t share a parent-offspring relationship) and other types of evolution. Unfortunately, the space of these models is huge and certain models with different topologies can give the same observed probabilities. In those cases, it is impossible to identify which graphical model our data is coming from. My research is focused on using methods from algebraic geometry to prove when we can and cannot identify the model our data is coming from and when we can  identify the parameters of the model. 

I have also been exploring ways we can use numerical methods to understand the space of positive real solutions of large polynomial systems. Biology provides many such systems where the positive real portion of the variety is extremely difficult to compute. However in many cases, this is also the only part of the variety which is physically realizable and thus of interest to biologists. I am interested in interested in answering questions about the numerical