I am currently working on the following projects:
Topology of Random Cubical Complexes: I am investigating various topological properties of random cubical complexes, in the spirit of Erdos', Renyi's, & Spencer's work on random graphs. In particular, I am interested in the Euler characteristic, low-order homology rank, homotopy type, and the existence of torsion in the homology & homotopy groups of these complexes. These complexes are constructed using different random processes assuming a uniform distribution of sampling. This is joint work with Jim Fowler & Matthew Kahle (Ohio State).
d-Regular Cubulations of Genus-g Surfaces: I am attempting to answer the following question: " For which g is there a d-regular cubulation of a genus-g surface, for some given d?" This is joint work with Jim Fowler (Ohio State).
Cohen-Lenstra Heuristics of Random Cohomology Groups: I am interested in knowing if certain cohomology groups of random spanning 2-trees are Cohen-Lenstra distributed. This is joint work with Jim Fowler (Ohio State), with brief consultation from Melanie Matchett Wood (Harvard, UC Berkeley).
Random Tilings and the Arctic Circle Theorem: I am currently studying random tilings of the plane and attempting to determine if the Arctic Circle Theorem for Aztec Diamonds can be applied to other classes of tilings called "Aztec Dragons" and "Aztec Castles." This is joint work with Gregg Musiker (Minnesota).