Some current and recent research projects:
- Tilings of a square
This applet demonstrates an example of a Monsky polynomial. If the given simplicial complex is drawn in the plane such that the boundary forms a square, then regardless of where the two internal vertices go, the areas of the triangles will satisfy the (irreducible) polynomial (A+C+E)2 – 4AC = (B+D+F)2 – 4DF. This is the Monsky polynomial associated to the simplicial complex. Move those vertices around and see for yourself!
I made the applet with GeoGebra. I do the math with Jamie Pommersheim.
- Homological and homotopical Dehn functions are different
This image shows part of the Cayley complex for the Baumslag-Solitar group ⟨ a,b | bab-1=a3 ⟩. The red loop is one of a sequence of loops that are used to show that this group has exponential Dehn function. The image, made in collaboration with Jason and Tammy Cantarella, appeared on the cover of the Proceedings of the National Academy of Sciences on November 26, 2013.A recent collaboration with Noel Brady, Pallavi Dani, Moon Duchin, and Robert Young (and sponsored by AIM) led to the article associated with the above image. We proved that homological and homotopical Dehn functions are different.
- Discretized configuration spaces
This is the discretized configuration space D2(K5). This means that it is the space of ordered pairs of points on the complete graph with 5 vertices. It is a 2-dimensional square complex with six squares meeting at each vertex. It is homeomorphic to a closed orientable surface of genus 6; it embeds in R3 as the boundary of a regular neighborhood of the graph K5. The image is based on a SketchUp model made by Michelle Chu. Over the years, beginning with my Ph.D. thesis, I have studied various aspects of discretized configuration spaces.