Many statistical models are fundamentally algebraic in that they can be described either parametrically or implicitly via polynomial. My research focuses on studying the mathematical and statistical properties of these algebraic statistical models. Major challenges are to elucidate how theoretical algebraic aspects of the models interact with applications. Currently, most problems I work on in this area are derived from applications including to phylogenetics, computational biology, causal inference, and disclosure limitation where most statistical models that are used are graphical models.