Research

My primary research interest is algebraic biology. This is an interdisciplinary field where tools from algebraic geometry are used to answer questions arising from biology. One example of this is algebraic statistics for phylogenetics or the study of reconstructing the evolutionary history of a set of related taxa from some observable data. I am interested in identifiability questions such as when is it possible to 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 flow 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. I use methods from algebraic geometry to prove when we can and cannot identify the graphical model our data is coming from and when we can  identify the parameters of the model. 

I am also interested in how we can use numerical methods to understand the space of positive solutions of large polynomial systems. Chemical reaction networks, population models from ecology are just some examples of phenomena which give systems of polynomial and rational ordinary differential equations where zeros correspond to the steady states. The nonnegative solutions correspond to realizable steady states and thus are the only steady states of interest. I am interested in studying how bifurcations of the steady states arise as the parameters change.