# Research - the questions that guide us

We work at the crossroads of geometry, combinatorics, and algebra.  For example, we relate configurations of geometric shapes (such as lines, points, and circles) with algebraic objects (such as polynomials) and combinatoric objects (such as polygons and directed graphs).  Specific projects include the following:

Families of algebraic varieties. Algebraic geometry studies the geometry of the zero sets of polynomials. Varying a polynomial's coefficients yields an algebro-geometric object. Therefore, our guiding questions are:  What possible zero sets (or isomorphism classes) occur by moving such coefficients toward a limit?  What is the geometry of the space parametrizing all zero sets and their limits?  We also use such projects to introduce students to moduli theory via the development of combinatorial and computational tools.

Polytopes, directed graphs, and families of linear maps. Given an acyclic-directed graph, if one considers all nonnegative flows on the edges subject to a boundary condition at the vertices, one arrives at a polytope parametrizing all such flows. Such a setting determines a classification problem because only finite polytopes exist for a given graph. We are interested in families of polytopes of dimensions larger than three and their connections to algebraic geometry.

## Software and computational tools

1. Variations of GIT quotients package. In joint work with Jesus Martinez-Garcia (University of Essex, UK), we developed software, in Python, that calculates the computational information required to describe the GIT quotients parametrizing a pair defined by hypersurface and a hyperplane.

2. Quivers and their thin sincere representations.  In work with Mary Barker (WashU), we developed a Macaulay2 package to describe their geometry and moduli spaces of thin-sincere representations of acyclic quivers i.e., the dimension vector is all equal to one.