My research is in the area of matroid structure theory. An important class of matroids is that of frame matroids. In a sense, these are finite geometries with graph-like structure. The complete graph on five vertices is shown in both the images at right and at left - the image in the side bar at left is its representation as a geometry: each edge of the graph is a point in the geometry. Every 3-cycle in the graph is a 3-point line in the geometry, every 4-cycle corresponds to four points that minimally span a plane; the edges of a forest in the graph are a set of affinely independent points in the geometry. Graphs provide a prototypical family of frame matroids.

Just as for graphs, families of matroids are often best understood by considering the unavoidable substructures (minors) all members of the family must or must not contain. For example, a graph may be embedded in the plane if and only if it does not contain the complete graph on five vertices or the complete bipartite graph with each part of size three.

Which structures are graphs? A classic theorem of Tutte lists five matroids that are minor-minimally not graphs. We thus completely understand which structures are fundamentally graphic, and which are not, and why.

The analogous question, which structures are frame matroids? has been the focus of my research. The universe of frame matroids is rich and wild, containing many natural structures in addition to those representable by graphs. It's complicated! We do not have a complete answer yet, but we are making progress.