My primary research interests are in topological combinatorics, especially applying techniques from discrete Morse theory to detect various properties about topological spaces generated by certain combinatorial structures. In particular, I am interested in calculating connectivity bounds, the Euler characteristic, dimensions of non-vanishing homology, and relevant cellular counting recursions for the spaces in question.
I was first introduced to discrete Morse theory during a Research Experience for Undergraduates at James Madison University (Harrisonburg, VA) in Summer 2011, where my research group worked primarily on the poset topology of pattern-avoiding permutation groups under the strong Bruhat order. Recently, I have been studying the homomorphism complexes generated by mapping chain posets into the Boolean algebras and extending these results to more general distributive lattices. Previously, I also worked with matching trees and the independence and matching complexes of small grid graphs along the lines of Bousquet-Mélou, Linusson, and Nevo.
“Homomorphism complexes of the Boolean algebras” (with B. Braun). Manuscript in preparation.
“Matching and independence complexes related to small grids” (with B. Braun). Electronic Journal of Combinatorics 24 (4), 2017, P4.18.
“Permutation pattern avoidance and the Catalan triangle” (with D. DeSantis, R. Field, B. Jones, R. Meissen, and J. Ziefle). Missouri Journal of Mathematical Sciences 25 (1), 2013, 50-60. Pre-print.